TheJustinSunPrize

Erdős问题 #650 迎来精确解：min(m, ⌈2√m⌉) #孙宇晨奖 聚焦数学经典问题的完整解决与完整Lean形式化验证。 解题者：Wouter van Doorn、Yanyang Li、 Quanyu Tang（@Vieta__jumping） Lean验证者：Wouter van Doorn 获奖记录👇 github.com/TheJustinSunPrize/awards/blob/main/awa… t.co

Open problem of the week Open problem of the week Take a fraction x in (0,1) with an odd denominator. Subtract 1/n, where n is the smallest odd number with 1/n ≤ x. Repeat on what is left. Does it always stop? Erdős Problem #282, Open in the #JustinSunPrize problem bank 👇 github.com/TheJustinSunPrize/awards/blob/main/pro…

Erdős Problem #650 has an exact answer: min(m, ⌈2√m⌉). The #JustinSunPrize register credits solution and Lean separately. Solution: Wouter van Doorn, Yanyang Li, Quanyu Tang (@Vieta__jumping) Lean formalization: Wouter van Doorn Award record 👇 github.com/TheJustinSunPrize/awards/blob/main/awa… t.co

本周未解问题： 取一个介于0和1之间、分母为奇数的分数x。从x中减去1/n，其中n是满足1/n ≤ x的最小正奇数，再对剩余的数重复这一操作。 这个过程一定会结束吗？ Erdős问题 #282，目前仍未解决，收录于 #孙宇晨奖 官方题库👇 github.com/TheJustinSunPrize/awards/blob/main/pro… t.co

How a #JustinSunPrize award is shared in the MathDB pilot: 50% Solution 30% Lean formalization 20% Exposition Each is reviewed separately. Solution and Exposition awards wait for a complete, verified Lean formalization of the problem. Pilot problems 👇 github.com/TheJustinSunPrize/awards/blob/main/pro… t.co

Looking for an open problem to work on? The #JustinSunPrize problem bank lists 1,072 problems, 677 currently marked Open. Justin Sun Prize problem IDs are not Erdős numbers, so read the statement and sources in each entry. Problem index 👇 github.com/TheJustinSunPrize/awards/blob/main/pro… t.co

The Littlewood conjecture in October 2026: what is proved, what changed in 2026, what is open What is proved, what changed in 2026, and what is still open about simultaneous approximation by a product of distances. This is an exposition with complete proofs of the elementary reductions (the facts that leave only badly approximable, linearly independent pairs) and of the step that turns a counterexample into a bounded orbit, plus a small, tested Python companion (littlewood.py) that computes certified continued fractions and certified Littlewood products. By MV2 of Munim, Inc. This article and the code were researched and written with AI agents working in MV2's name. Every statement below is either proved here or cited to a source. Section 10 separates what is proved from what is open, and section 11 lists what is new here. Corrections are welcome by issue or pull request. Version 1.0, 2026-10-14. Text CC BY 4.0, code Apache-2.0. Related problem records: Justin Sun Prize JSP-000016 (MathDB #316099, "Littlewood conjecture"), Wikipedia "Littlewood conjecture". --- 1. The question Write ‖t‖ for the distance from a real number t to the nearest integer. Littlewood's conjecture (posed by J. E. Littlewood around 1930) says: > For all real numbers α and β, liminf_{q→∞} q · ‖qα‖ · ‖qβ‖ = 0. Dirichlet's theorem gives, for one number, infinitely many q with q‖qα‖ < 1, and for two numbers infinitely many q with ‖qα‖, ‖qβ‖ < q^{−1/2}. Both give a product q‖qα‖‖qβ‖ that is merely bounded. The conjecture asks for more: the product of the two errors, scaled by q, can be made as small as we like. Two equivalent forms are used below. - The infimum form. Put m(α, β) = inf_{q ≥ 1} q‖qα‖‖qβ‖. The conjecture is equivalent to m(α, β) = 0 for all pairs (Lemma 1), i.e. to m_LC = sup_{α,β} m(α, β) being 0. - Products of linear forms. m(α, β) = inf |x (xα − y)(xβ − z)| over integers x ≠ 0, y, z (take y, z nearest to xα, xβ). So the conjecture asks whether this product of three linear forms in (x, y, z) takes values arbitrarily close to 0 at integer points with x ≠ 0. 2. Elementary reductions, with proofs Lemma 1 (infimum versus liminf). liminf q‖qα‖‖qβ‖ = 0 if and only if m(α, β) = 0. Proof. One direction is clear. Conversely, if some term is 0 then α or β is rational, and then infinitely many terms are 0 (multiples of the denominator). If no term is 0, all terms are positive, so an infimum of 0 is not attained and is approached by infinitely many q. ∎ Continued fractions. Let α = [a₀; a₁, a₂, ...] be irrational, with convergents pₖ/qₖ (q₋₁ = 0, q₀ = 1, qₖ₊₁ = aₖ₊₁qₖ + qₖ₋₁). Writing α = [a₀; ..., aₖ, xₖ₊₁] with complete quotient xₖ₊₁ ∈ (aₖ₊₁, aₖ₊₁ + 1), the identity qₖα − pₖ = ±1/(xₖ₊₁qₖ + qₖ₋₁) gives, for k ≥ 1, 1 / (qₖ₊₁ + qₖ) < ‖qₖα‖ < 1 / qₖ₊₁. (Legendre's bounds) α is badly approximable (α ∈ Bad) if inf_q q‖qα‖ > 0. By the bounds, qₖ‖qₖα‖ < qₖ/qₖ₊₁ < 1/aₖ₊₁, so α ∉ Bad when the partial quotients are unbounded. (Conversely, bounded partial quotients imply α ∈ Bad, because the best approximations are convergents; this is classical.) Lemma 2 (one coordinate suffices). If α is rational or α ∉ Bad, the conjecture holds for (α, β) for every β. Proof. Rational α: shown in Lemma 1. Otherwise inf_q q‖qα‖ = 0 and ‖qβ‖ ≤ 1/2, so m(α, β) ≤ ½ inf_q q‖qα‖ = 0; apply Lemma 1. ∎ Lemma 3 (linear dependence, with an explicit rate). If a + bα + cβ = 0 for integers a, b, c not all zero, the conjecture holds for (α, β). More precisely, if c ≠ 0 and α is irrational, then for every convergent denominator qₖ of α (k ≥ 1) the integer n = |c|qₖ satisfies n ‖nα‖ ‖nβ‖ ≤ c² |b| qₖ / qₖ₊₁² < c² |b| / qₖ₊₁. Proof. If c = 0 then α is rational (b ≠ 0), and Lemma 2 applies; if α is rational, likewise. Otherwise ‖nα‖ = ‖|c|·qₖα‖ ≤ |c|·‖qₖα‖ < |c|/qₖ₊₁, using ‖mt‖ ≤ |m|·‖t‖ for integers m. And nβ = ±qₖ(cβ) = ∓qₖ(a + bα), so ‖nβ‖ = ‖qₖbα‖ ≤ |b|·‖qₖα‖ < |b|/qₖ₊₁. Multiply by n = |c|qₖ. ∎ The test suite checks this for β = (1 + 2√2)/3, α = √2: with qₖ = 2378, qₖ₊₁ = 5741 and n = 7134 the bound is 18 · 2378 / 5741² < 0.0013, and the computed value is about 0.00095. Lemma 4 (a universal bound). m(α, β) < 1/(2√5) = 0.2236... for every pair. Proof. By Hurwitz's theorem every irrational α has infinitely many q with q‖qα‖ < 1/√5 (Borel: one of any three consecutive convergents works); multiply by ‖qβ‖ ≤ 1/2. Rational α gives m = 0. ∎ The real question is how small m(α, β) can be made uniformly: Badziahin (2016) proved m_LC < 1/19 and Matala-Aho, Törmä and Vapa (2026) proved m_LC < 1/26 (section 7). A proof that m_LC = 0 is the conjecture. Lemma 5 (quadratic irrationals are badly approximable). If α is a real root of an irreducible f(x) = Ax² + Bx + C ∈ ℤ[x], then α ∈ Bad. Proof. For integers p, q ≥ 1, q²f(p/q) is a nonzero integer, so |f(p/q)| ≥ 1/q². If |α − p/q| ≤ 1, then |f(p/q)| = |A|·|p/q − α|·|p/q − α'| ≤ |A|(|α − α'| + 1)·|α − p/q|. Hence q²|α − p/q| ≥ 1/(|A|(|α − α'| + 1)) for all p/q close to α, and q‖qα‖ is bounded below. ∎ Where this leaves the problem. A counterexample must be a pair of badly approximable numbers α, β with 1, α, β linearly independent over ℚ. (√2, √3) is such a pair (Lemma 5; 1, √2, √3 are independent), and the conjecture is not known for it (Gowers 2009; we found no later result settling it). By Jarník, Bad has Hausdorff dimension 1, so these reductions alone leave a set of dimension 2. 3. Almost every pair, and the algebraic pairs - Almost every pair. Bad has Lebesgue measure 0 (Khintchine 1924; Borel–Bernstein), so by Lemma 2 the possible counterexamples form a null set. Gallagher (1962) proved much more: for almost every (α, β), liminf q (log q)² ‖qα‖‖qβ‖ = 0. Chow and Yang (Invent. Math. 2024) proved Gallagher-type statements on planar lines, using effective equidistribution. - Same cubic field. Cassels and Swinnerton-Dyer (1955) proved the conjecture when 1, α, β span a real cubic field, by proving the dual statement inf_{m,n≠0} |mn|·‖mα + nβ‖ = 0 and transferring it. Peck (1961) sharpened this: for a basis 1, α₁, ..., α_d of a real number field of degree d + 1, liminf q log q ‖qα₁‖...‖qα_d‖ < ∞. In September 2026 Dhanda, Flynn and Haynes gave a geometric proof of Peck's theorem from one application of Minkowski's convex body theorem (arXiv 2609.29469), and Bugeaud and de Mathan proved refinements toward Peck's conjecture on splitting the logarithmic saving unequally between coordinates (arXiv 2609.29360). - Two different quadratic fields, such as (√2, √3), are not covered: they live in the quartic field ℚ(√2, √3), and Peck-type results for its natural basis 1, √2, √3, √6 involve the third number √6. - A related open problem. No algebraic number of degree ≥ 3 is known to be badly approximable, and none is known not to be. The companion code certifies the first 41 partial quotients of 2^{1/3}, [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1] (OEIS A002945); the 534 is a famous surprise. If cubic irrationals are not badly approximable, as is widely expected, the Cassels–Swinnerton-Dyer theorem follows immediately from Lemma 2. 4. Dynamics: why the exceptional set is small Attach to (α, β) the lattice Λ = { (x, xα − y, xβ − z) : x, y, z ∈ ℤ } ⊂ ℝ³ (covolume 1, so a point of X₃ = SL(3, ℝ)/SL(3, ℤ)), and the diagonal matrices a_{s,t} = diag(e^{−s−t}, e^{s}, e^{t}) with s, t ≥ 0. They preserve the product of the three coordinates. Lemma 6 (a counterexample gives a bounded orbit). If m(α, β) = δ > 0, then every nonzero vector of every lattice a_{s,t}Λ (s, t ≥ 0) has sup-norm at least min(1, δ^{1/3}). By Mahler's compactness criterion the orbit {a_{s,t}Λ : s, t ≥ 0} is then bounded in X₃. Proof. Take a nonzero v = a_{s,t}(x, xα − y, xβ − z). If x ≠ 0, the product of the coordinates of v is x(xα − y)(xβ − z), of absolute value ≥ δ, so its largest coordinate is ≥ δ^{1/3}. If x = 0, then v = (0, −e^{s}y, −e^{t}z) with (y, z) ≠ (0, 0) and s, t ≥ 0, so ‖v‖ ≥ 1. ∎ So the conjecture would follow from: no such orbit is bounded. This is the case d = 3 of conjectures of Cassels–Swinnerton-Dyer (products of d ≥ 3 linear forms) and Margulis (bounded orbits of the full diagonal group are compact). Maucourant (2010) showed that the analogous orbit-closure statement fails for diagonal subgroups of SL(d, ℝ) with d ≥ 6; this does not affect Littlewood. Einsiedler, Katok and Lindenstrauss (Annals 2006) classified the invariant measures of positive entropy for the diagonal group on X₃ and deduced: the set of pairs (α, β) for which the conjecture fails has Hausdorff dimension 0; in fact it is a countable union of compact sets of box dimension 0. This is still the strongest general result (Venkatesh's 2008 survey explains the proof). It does not show the set is empty. Einsiedler and Kleinbock (2007) proved the same dimension statement for the p-adic version (section 8). 5. Explicit pairs and limits of strengthening - Pollington–Velani (Acta Math. 2000): for every α ∈ Bad, there is a set of β ∈ Bad of Hausdorff dimension 1 with q log q ‖qα‖‖qβ‖ ≤ 1 for infinitely many q. - de Mathan (2003) and Adamczewski–Bugeaud (2006): explicit pairs of badly approximable numbers, with 1, α, β independent, that satisfy the conjecture; for each α with bounded partial quotients, Adamczewski and Bugeaud construct uncountably many such β. - How much can be gained over Littlewood? Gallagher's (log q)² holds almost everywhere, but Bugeaud and Moshchevitin (2011) showed that the pairs with liminf q (log q)² ‖qα‖‖qβ‖ > 0 have full Hausdorff dimension, and Badziahin (2013) did the same with q log q log log q. Fregoli and Kleinbock (J. Number Theory, 2026) extended Badziahin's result to ℝ^d with (log q)^{d−1} log log q. - Inhomogeneous versions. Shapira (Annals 2011): for almost every (α, β), for all shifts (γ, δ), liminf |n|‖nα − γ‖‖nβ − δ‖ = 0; this also holds for pairs from totally real cubic fields. Further results on fibres by Chow–Zafeiropoulos (2021), Chow–Technau (2024) and Stefanescu (2025). 6. Products of three linear forms in a picture For a fixed pair, the companion records every q at which q‖qα‖‖qβ‖ reaches a new minimum (python littlewood.py records sqrt2 sqrt3 1000000). A conjecture-sized question becomes concrete: does this record sequence tend to 0 for (√2, √3)? A computation can only ever show values getting small; it cannot prove the liminf is 0, and it cannot exhibit a counterexample. One value that can be checked by hand: at q = 41, 41√2 = 57.98275..., 41√3 = 71.01408..., 41 · 0.0172439 · 0.0140831 = 0.0099568... already below the universal bound 1/26 of section 7. (python littlewood.py value sqrt2 sqrt3 41 prints a certified interval of width below 10⁻³⁰.) 7. What has been computed - Universal bounds on m_LC. Badziahin (Exp. Math. 2016) gave an algorithm that certifies inequalities m_LC < ε, and proved m_LC < 1/19 (and an analogous 2-adic bound of 1/9). Matala-Aho, Törmä and Vapa (Exp. Math., online 2026-04-02) gave a new algorithm based on the continued fractions of α and β and proved inf_q q‖qα‖‖qβ‖ < 1/26 for all real α, β. These are upper bounds for the worst pair; the conjecture says the worst pair still gives 0. - p-adic numerics. Badziahin's "Estimating lower limit in the p-adic Littlewood conjecture" (arXiv 2506.16860, 2025) and Vitorino–Vucusic (Mathematika 2026) give further bounds for the 2-adic case. 8. The 2025–2026 developments (a) The uniform Littlewood conjecture is false. Bandi, Fregoli and Kleinbock (arXiv 2504.02258, 2025) proposed a uniform strengthening: for every pair, lim_{Q→∞} Q · min_{1≤q≤Q} ‖qα‖‖qβ‖ = 0, which they knew to hold for almost every pair. Since q ≤ Q, this implies Littlewood's conjecture; it asks that every large Q has a good q ≤ Q, not just infinitely many. In 2026: - Schleischitz (arXiv 2603.12611, March) disproved it: the counterexamples form a residual set (a dense G_δ). The disproof is semi-constructive, using effective results toward Zaremba's conjecture (Bourgain–Kontorovich). He also disproved a uniform p-adic version. - Moshchevitin (arXiv 2605.26188, May) gave a short proof of a counterexample using properties of Fibonacci numbers. - Shulga (arXiv 2608.25059, August) proved the counterexamples with badly approximable first coordinate form a set of Hausdorff dimension at least 3/2. - Neckrasov, Wu and Yang (arXiv 2608.24401, August–September) proved the set of counterexamples is hyperplane absolute winning, so it has full Hausdorff dimension 2. This does not affect the classical conjecture, whose exceptional set has dimension 0 (section 4). The contrast is the point: the uniform statement fails on a big set; the classical one can fail only on a set of dimension 0. python littlewood.py uniform X Y Q prints the uniform quantity. (b) Function fields: the analogy breaks in odd characteristic. De Mathan and Teulié also proposed an analogue of the p-adic conjecture (part (c)) for Laurent series over a finite field 𝔽_q, with an irreducible polynomial P(t) in place of p: the t-adic and P(t)-adic Littlewood conjectures. Adiceam, Nesharim and Lunnon (Duke 2021) found an explicit counterexample over 𝔽₃. Garrett and Robertson (Math. Comp. 2026) found computer-assisted counterexamples to the P(t)-adic version over small fields (characteristic 5, and the characteristic-3 construction in characteristics 7 and 11); Robertson (Mathematika 2026) gave a transference from t to any irreducible P(t) using number walls. Lai and Sprang (arXiv 2606.00633, May 2026) proved that the P(t)-adic Littlewood conjecture fails for every irreducible P(t) over every ground field of odd characteristic. Characteristic 2 is open. Badziahin, Pavlenkov and Zorin (arXiv 2608.22078, August 2026) showed that in the t-adic case a non-empty exceptional set has infinite 1/log(1/r)-Hausdorff measure, refining the Lai–Sprang construction for odd q. None of this implies anything about real numbers. (c) The p-adic Littlewood conjecture (de Mathan–Teulié 2004): for every real α and prime p, inf_q q·|q|_p·‖qα‖ = 0. Proved for quadratic irrationals (de Mathan–Teulié); exceptional set of dimension 0 (Einsiedler–Kleinbock 2007); still open. Badziahin, Pavlenkov and Zorin (2026) showed that if the exceptional set is non-empty, it has positive Hausdorff measure for the gauge 1/log(1/r), so it has the cardinality of the continuum. python littlewood.py mixed X P Q records q|q|_P‖qX‖. 9. Status as of 2026-10-14 - Open. We found no 2025–2026 result proving or disproving the classical conjecture. The most recent papers on it (Shulga, August 2026; Neckrasov–Wu–Yang, September 2026) describe it as open, with the dimension-0 theorem of Einsiedler–Katok–Lindenstrauss as the best general result; Wikipedia's article (edited 2026-10-05) agrees. The Justin Sun Prize MathDB pilot register lists JSP-000016 with no solution, Lean or exposition entry. We found no claimed proof on arXiv in 2025–2026. - What moved in 2026: the uniform version was disproved (with full-dimensional counterexamples), the P(t)-adic analogue was disproved in every odd characteristic, the universal bound improved from 1/19 to 1/26, Badziahin's logarithmic obstruction was extended to higher dimensions, and Peck's theorem got a geometric proof and refinements. 10. Proved versus open, as of 2026-10-14 Proved: 1. The conjecture holds when α or β is rational or not badly approximable, and when 1, α, β are linearly dependent over ℚ (Lemmas 2 and 3). 2. m(α, β) < 1/26 for every pair (Matala-Aho–Törmä–Vapa 2026; earlier 1/19, Badziahin 2016; 1/(2√5) by Hurwitz, Lemma 4). 3. It holds for almost every pair, with the factor (log q)² (Gallagher 1962), and for pairs spanning a real cubic field with 1, with the factor log q (Cassels–Swinnerton-Dyer 1955; Peck 1961). 4. The exceptional set has Hausdorff dimension 0 (Einsiedler–Katok–Lindenstrauss 2006); same for the p-adic version (Einsiedler–Kleinbock 2007). 5. For each α ∈ Bad, a dimension-1 set of β ∈ Bad satisfies it (Pollington–Velani 2000); explicit independent badly approximable pairs satisfy it (de Mathan 2003; Adamczewski–Bugeaud 2006). 6. The uniform Littlewood conjecture is false, with counterexamples of full dimension (2026). 7. The P(t)-adic analogue is false over every field of odd characteristic (Lai–Sprang 2026, building on Adiceam–Nesharim–Lunnon 2021 and Garrett–Robertson 2026). Open: 1. Littlewood's conjecture itself: is m(α, β) = 0 for every pair? Equivalently, is m_LC = 0? 2. Any specific independent pair of algebraic numbers from different quadratic fields, starting with (√2, √3). 3. The p-adic Littlewood conjecture (de Mathan–Teulié), and the t-adic one in characteristic 2. 4. The Cassels–Swinnerton-Dyer and Margulis conjectures on diagonal orbits in X₃. 5. Whether any algebraic number of degree ≥ 3 is badly approximable. 11. What is original here - Self-contained proofs of the reductions (Lemmas 1–5), including an explicit rate c²|b|qₖ/qₖ₊₁² in the linearly dependent case, and of the step from a counterexample to a bounded diagonal orbit (Lemma 6). - A certified continued fraction method from an exact floor oracle: if N·x is available as ⌊N x⌋ in integers, the reals whose floor algorithm starts with a₀, ..., a_{j−1} form an interval, so the terms shared by the expansions of ⌊2ᵇx⌋/2ᵇ and (⌊2ᵇx⌋ + 1)/2ᵇ are proved terms of x. With integer k-th roots this certifies, for example, 41 partial quotients of 2^{1/3}, including the 534. - Certified Littlewood products: since ‖·‖ is 1-Lipschitz and qx lies in [qA/S, (qA + q)/S) for A = ⌊Sx⌋, each value q‖qα‖‖qβ‖ comes with a proved interval; with S = 2²⁵⁶ the width is far below 10⁻³⁰ for q in the millions. - littlewood.py and test_littlewood.py: exact reals ((a + b√d)/c and (p/q)^{1/k}), certified continued fractions and convergents, certified distances, record scans of q‖qα‖‖qβ‖, the uniform quantity of section 8(a), and the p-adic quantity of section 8(c), in plain Python with no dependencies. The tests check Legendre's bounds and Borel's theorem on convergents, Lemma 3 on a dependent pair, the value at q = 41 for (√2, √3), and the cube root of 2 against OEIS A002945. Nothing here claims progress on the conjecture itself. 12. Using the companion code python littlewood.py cf cbrt2 41 # certified partial quotients of 2^(1/3) python littlewood.py records sqrt2 sqrt3 100000 # new minima of q ||q sqrt2|| ||q sqrt3|| python littlewood.py value sqrt2 sqrt3 41 # certified interval for one value python littlewood.py uniform sqrt2 sqrt3 100000 # T * min_{q<=T} ||q x|| ||q y||, T = 1, 2, 4, ... python littlewood.py mixed sqrt2 2 100000 # new minima of q |q|_2 ||q sqrt2|| python -m unittest test_littlewood # or: python test_littlewood.py Numbers: sqrt2, sqrt(7), cbrt2, 2^1/3, 2^2/3, golden, and surd:a,b,d,c for (a + b√d)/c. Python 3.6+, no dependencies. Sources - J. W. S. Cassels, H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms, Phil. Trans. Roy. Soc. A 248 (1955) 73–96. doi:10.1098/rsta.1955.0010 - L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. AMS 67 (1961) 197–201. - P. X. Gallagher, Metric simultaneous diophantine approximation, J. London Math. Soc. 37 (1962) 387–390. doi:10.1112/jlms/s1-37.1.387 - A. D. Pollington, S. L. Velani, On a problem in simultaneous diophantine approximation: Littlewood's conjecture, Acta Math. 185 (2000) 287–306. doi:10.1007/BF02392812 - G. A. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, AMS, 2000, 161–174. - B. de Mathan, Conjecture de Littlewood et récurrences linéaires, J. Théor. Nombres Bordeaux 15 (2003) 249–266. doi:10.5802/jtnb.401 - B. de Mathan, O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004) 229–245. doi:10.1007/s00605-003-0199-y - B. Adamczewski, Y. Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc. 73 (2006) 355–366. doi:10.1112/S0024610706022617 - M. Einsiedler, A. Katok, E. 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Lehmer's problem in October 2026: what is proved, what has been computed, what is open What is proved, what has been computed, and what is still open about the smallest Mahler measure. This is an exposition with complete proofs of the elementary theory (including Kronecker's theorem, proved in a form that doubles as an exact algorithm) and a small, tested Python companion (lehmer.py) that measures polynomials, certifies when the measure is exactly 1, and re-runs the small searches behind the statements below. By MV2 of Munim, Inc. This article and the code were researched and written with AI agents working in MV2's name. Every statement below is either proved here or cited to a source; section 10 separates what is proved from what is open, and section 11 lists what is new here. Corrections are welcome by issue or pull request. Version 1.0, 2026-10-14. Text CC BY 4.0, code Apache-2.0. Related problem records: Justin Sun Prize JSP-000025 (MathDB #315901, "Lehmer–Mahler measure problem"), Wikipedia "Lehmer's conjecture", Mossinghoff's tables of small Mahler measures. --- 1. The question For a nonzero polynomial with integer coefficients P(x) = aₙ xⁿ + ... + a₀ = aₙ (x − α₁)(x − α₂) ... (x − αₙ), the Mahler measure is M(P) = |aₙ| · ∏ max(1, |αᵢ|). For an algebraic number α, M(α) means the measure of its minimal polynomial over ℤ. M(P) ≥ 1 always (|aₙ| ≥ 1), and M(P) = 1 exactly when P is ± a power of x times a product of cyclotomic polynomials (Kronecker, section 3). Lehmer's problem (Lehmer 1933) asks whether the measure can come arbitrarily close to 1 from above: > Lehmer's conjecture. There is a constant μ > 1 such that every P ∈ ℤ[x] has M(P) = 1 or M(P) ≥ μ. The smallest measure above 1 that anyone has found is Lehmer's own example from 1933: L(x) = x¹⁰ + x⁹ − x⁷ − x⁶ − x⁵ − x⁴ − x³ + x + 1, M(L) = 1.17628081825991750... and it is widely expected that μ = M(L) is the best constant. In 93 years nobody has found a smaller measure above 1, and nobody has proved any constant μ > 1. 2. The Mahler measure: basic facts, with proofs Write P*(x) = xⁿ P(1/x) for the reciprocal (coefficients reversed). P is called reciprocal when P* = ±P. (a) Multiplicativity. M(PQ) = M(P) M(Q): leading coefficients multiply and the root lists concatenate. (b) Reversal. M(P*) = M(P) when P(0) ≠ 0: the roots of P* are 1/αᵢ, its leading coefficient is a₀ = aₙ ∏(−αᵢ), and |a₀| ∏ max(1, 1/|αᵢ|) = |aₙ| ∏ |αᵢ| max(1, 1/|αᵢ|) = |aₙ| ∏ max(|αᵢ|, 1). (c) Jensen's formula. log M(P) = ∫₀¹ log |P(e^{2πit})| dt. This is Jensen's formula for the disc applied to each factor: ∫₀¹ log |e^{2πit} − α| dt = log max(1, |α|). (Mahler used this integral to extend the measure to polynomials in several variables; section 8.) (d) Coefficient bound. For every j, |aⱼ| ≤ C(n, j) · M(P). Proof. aⱼ = ±aₙ eₙ₋ⱼ(α₁, ..., αₙ), and the elementary symmetric function eₖ is a sum of C(n, k) products of k roots, each of absolute value at most ∏ max(1, |αᵢ|). ∎ (e) Landau's inequality. M(P) ≤ ‖P‖₂ = (Σ |aⱼ|²)^{1/2}. Proof. By (c) and concavity of log, log M(P) = ½ ∫ log |P|² ≤ ½ log ∫ |P|² = log ‖P‖₂, the last step by Parseval. ∎ (f) Finiteness (Northcott-type). For fixed n and B, only finitely many P ∈ ℤ[x] of degree ≤ n have M(P) ≤ B, by (d). So for each n the smallest measure above 1 among polynomials of degree ≤ n, call it mₙ, exists, and m₁ ≥ m₂ ≥ ... . Lehmer's question is exactly whether inf mₙ > 1. The difficulty is entirely in the growth of the degree. (g) Root squaring. Write P(x) = E(x²) + x O(x²) and put G(P)(y) = (−1)ⁿ (E(y)² − y O(y)²). Then G(P)(x²) = (−1)ⁿ P(x) P(−x) = aₙ² ∏ (x² − αᵢ²), so G(P) is an integer polynomial of the same degree with roots αᵢ² and leading coefficient aₙ², and therefore M(G(P)) = M(P)². This is Graeffe's classical root-squaring step; everything in section 3 and the certified bounds in section 11 are built on it. 3. Kronecker's theorem, proved as an algorithm Theorem (Kronecker 1857). Let P ∈ ℤ[x] be monic with P(0) ≠ 0, and suppose every root of P lies in the closed unit disc. Then every root of P is a root of unity. Proof. Let Pₖ = G(Pₖ₋₁), P₀ = P (section 2(g)). Each Pₖ is monic of degree n with integer coefficients and roots αᵢ^{2ᵏ}, all in the closed unit disc, so M(Pₖ) = 1 and by 2(d) the coefficient of xʲ in Pₖ is at most C(n, j) in absolute value. There are only finitely many such polynomials, so Pₐ = P_b for some a < b. Put c = b − a and βᵢ = αᵢ^{2ᵃ}. Equality of Pₐ and P_b means the multiset {βᵢ} equals the multiset {βᵢ^{2ᶜ}}, so there is a permutation σ with β_{σ(i)} = βᵢ^{2ᶜ}. If σʳ is the identity, then βᵢ^{2^{cr}} = βᵢ, and βᵢ ≠ 0, so βᵢ is a root of unity; hence so is αᵢ. ∎ Corollary. For P ∈ ℤ[x], M(P) = 1 if and only if P = ±xᵏ ∏ Φₘ^{eₘ}. Proof. M(P) ≥ |aₙ| ≥ 1, so M(P) = 1 forces |aₙ| = 1 and all roots in the closed disc. After removing the factor xᵏ, Kronecker's theorem makes every root a root of unity, and the minimal polynomial of a primitive m-th root of unity is the cyclotomic polynomial Φₘ (irreducibility of Φₘ over ℚ). The converse is clear. ∎ The proof is an algorithm. lehmer.is_kronecker runs it exactly, with Python integers: - If M(P) = 1, then P = ±∏ Φₘ^{eₘ}, and root squaring acts on cyclotomic factors as Φₘ ↦ Φₘ for odd m, Φ₂ₘ ↦ Φₘ for odd m, and Φₘ ↦ Φₘ/₂² when 4 | m (squaring permutes, halves, or folds the primitive roots two-to-one). So the sequence Pₖ reaches a fixed point after at most v + 1 steps, where 2ᵛ is the largest power of 2 dividing an order of a root. Conversely, a fixed point G(Q) = Q forces every root of Q onto the unit circle (the largest and smallest root moduli r satisfy r² = r), so M(P) = 1. - If M(P) > 1, then M(Pₖ) = M(P)^{2ᵏ}. While every coefficient obeys |cⱼ| ≤ C(n, j), Landau's inequality gives M(Pₖ) ≤ (Σ C(n, j)²)^{1/2} = C(2n, n)^{1/2} < 2ⁿ, so some coefficient breaks its binomial bound as soon as 2ᵏ log M(P) ≥ n log 2. Either way the test stops after a handful of squarings, and its answer is a proof. For Lehmer's polynomial it certifies M(L) > 1; for Φ₁₀₅ (the first cyclotomic polynomial with a coefficient −2) it certifies M = 1. The test suite checks that this decision agrees with explicit cyclotomic factorisation on every polynomial of degree ≤ 4 with coefficients in {−1, 0, 1}. 4. Where the question came from Lehmer (1933) was looking for large primes. For monic P with roots αᵢ he studied Δₖ = ∏ (αᵢᵏ − 1), an integer (a resultant of P and xᵏ − 1) that generalises the Mersenne numbers: for P = x − 2, Δₖ = 2ᵏ − 1. When no root lies on the unit circle, |αᵏ − 1|^{1/k} → max(1, |α|), so |Δₖ|^{1/k} → M(P). Primes are easier to find among numbers that grow slowly, so Lehmer wanted M(P) close to 1, and asked how close it can get. (When P has roots on the unit circle that are not roots of unity the same limit holds, as Lind noted, via Gelfond's or Baker's bounds on linear forms in logarithms.) lehmer.pierce computes Δₖ exactly as the determinant of multiplication by P on ℤ[x]/(xᵏ − 1), a circulant matrix whose eigenvalues are P(ζ) for ζᵏ = 1; the tests check it against 2ᵏ − 1 and against the Lucas-number formula Δₖ = (−1)ᵏ − Lₖ + 1 for x² − x − 1. 5. Salem and Pisot numbers M(L) is the real root λ = 1.17628... of L. Its conjugates are 1/λ and eight numbers of absolute value exactly 1. A real algebraic integer > 1 whose other conjugates lie in the closed unit disc, with at least one on the circle, is a Salem number; if all the others are strictly inside, it is a Pisot number. If α is either, M(α) = α. So λ is a Salem number, and Lehmer's conjecture implies that Salem numbers are bounded away from 1. (The reverse implication is not known.) The smallest Pisot number is θ₀ = 1.32471795724..., the real root of x³ − x − 1 (Siegel 1944). Every Pisot number is a limit of Salem numbers (Salem). The smallest known Salem number is λ; 47 Salem numbers below 1.3 are known, and the list is certified complete up to degree 44 (Flammang–Grandcolas–Rhin to degree 40; Mossinghoff–Rhin–Wu 2008 to degree 44). A 2025 search by Sac-Épée recovered all 47 and found no new ones below 1.3 (it did find 10 new Salem numbers below 49/37, of degrees 26 to 44). 6. What is proved: partial results Non-reciprocal polynomials are settled. If P ∈ ℤ[x], P(0) ≠ 0 and P is not reciprocal, then M(P) ≥ θ₀ = M(x³ − x − 1) = 1.3247... (Smyth 1971; Breusch 1951 had 1.1796...) and the bound is sharp. Smyth also showed θ₀ is isolated: apart from α = (±θ₀)^{±1/k}, a non-reciprocal algebraic integer has M(α) > θ₀ + 10⁻⁴. (Why reciprocity is the issue: if P is non-reciprocal, some irreducible factor f is; f is not cyclotomic, because cyclotomic polynomials are reciprocal, so Smyth's theorem applies to f, and M(P) ≥ M(f).) So Lehmer's problem is a problem about reciprocal polynomials, equivalently about algebraic integers α conjugate to 1/α. Bounds that shrink with the degree d of α (α not zero or a root of unity): | Result | Bound | |---|---| | Blanksby–Montgomery 1971; Stewart 1978 | log M(α) ≥ C / (d log d) | | Dobrowolski 1979 | log M(α) ≥ C (log log d / log d)³, with C = 1/1200, and C close to 1 for large d | | Cantor–Straus 1982; Louboutin 1983 | the same, with asymptotic constant 2, then 9/4 | | Voutier 1996 | C = 1/4, effective for all d ≥ 2 | Dobrowolski's method uses the fact that the resultant of the minimal polynomials of α and αᵖ is divisible by pᵈ for every prime p. Since 1979 only the constant has improved, not the shape (log log d / log d)³; Lehmer's conjecture asks for a lower bound on log M(α) that does not decrease with d at all. Special classes where Lehmer's conjecture is proved: - Totally real α (all conjugates real, α ≠ 0, ±1): M(α) ≥ ((1 + √5)/2)^{d/2}, with equality for the golden ratio (Schinzel 1973). - α in an abelian extension of ℚ (α not zero or a root of unity): the absolute height h(α) = log M(α)/d is at least (log 5)/12 (Amoroso–Dvornicich 2000), so M(α) ≥ 5^{d/12}. - Odd coefficients. If f = Σ_{k<n} aₖ xᵏ has all aₖ odd and no cyclotomic factor, then log M(f) ≥ (log 5 / 4)(1 − 1/n) (Borwein–Dobrowolski–Mossinghoff 2007). If such f is also non-reciprocal, M(f) ≥ M(x² − x − 1) = 1.618... (Borwein–Hare–Mossinghoff 2004). - Integer symmetric matrices. For the reciprocal polynomial attached to the characteristic polynomial of a non-cyclotomic integer symmetric matrix A, M(A) ≥ λ, Lehmer's number exactly (McKee–Smyth 2012; graphs earlier, 2005). This does not settle the general case, because some small Salem numbers are not of this form (for example M(z¹⁴ − z¹² + z⁷ − z² + 1) = 1.2026...). A weaker conjecture that is now proved. The house of an algebraic integer α is the largest absolute value of its conjugates. Since M(α) ≤ house(α)ᵈ, Lehmer's conjecture would give house(α) ≥ μ^{1/d} ≥ 1 + (log μ)/d, which is the Schinzel–Zassenhaus conjecture (1965). Dimitrov (arXiv 2019) proved it: house(α) ≥ 2^{1/(4d)} for every nonzero algebraic integer α of degree d that is not a root of unity. The implication does not reverse, so Lehmer's problem is untouched by it. 7. What has been computed - Boyd found every irreducible non-cyclotomic polynomial with M < 1.3 up to degree 20; Mossinghoff (1998) extended this to degree 24 and found none with measure below M(L). - Flammang, Rhin and Sac-Épée found all M(α) < θ₀ up to degree 36. - Mossinghoff, Rhin and Wu (2008) computed every primitive irreducible non-cyclotomic polynomial with M < 1.3 up to degree 44, and determined the minimal measure in each even degree up to 54. The smallest of these is M(L). Odd degrees are covered by Smyth, because an irreducible reciprocal polynomial of degree ≥ 2 has even degree (its roots pair off as α, 1/α, and a fixed point would be ±1). So every algebraic number of degree ≤ 54 that is not zero or a root of unity has M(α) ≥ M(L). - Mossinghoff's public list of known polynomials with M < 1.3 runs to degree 72 and is complete through degree 44. The companion code reproduces a tiny corner of this: among the 243 monic reciprocal polynomials of degree 10 with coefficients in {−1, 0, 1}, the smallest measure above 1 is M(L), attained by L (python lehmer.py search 5); and every monic non-reciprocal polynomial of degree ≤ 6 with coefficients in {−1, 0, 1} and nonzero constant term has M ≥ θ₀, with equality attained (python lehmer.py smyth 6). Both are asserted in the test suite, with M = 1 decided exactly by the Kronecker test of section 3, not by a floating-point threshold. 8. Why the problem matters outside number theory - Knots. L(−x) is the Alexander polynomial of the (−2, 3, 7) pretzel knot (it appears in Reidemeister's 1932 book, a year before Lehmer's paper). Hironaka showed this knot minimises the measure in a family of pretzel links. - Coxeter groups. λ is the spectral radius of the Coxeter element of E₁₀, and McMullen (2002) proved that for Coxeter elements the spectral radius is either 1 or at least λ: Lehmer's conjecture holds in that setting. - Dynamics on surfaces. An automorphism of a compact complex surface with positive entropy has entropy at least log λ (McMullen 2007), and the bound is attained on a rational surface (2007) and on a projective K3 surface (McMullen 2016; explicit equations by Brandhorst–Elkies). - Ergodic theory. The entropy of an ergodic automorphism of a compact abelian group, when finite, is a logarithmic Mahler measure. Lind (1977) showed that the set of possible entropies is either all of (0, ∞] or countable, according to how Lehmer's problem is resolved. - Several variables. Jensen's formula extends M to polynomials in several variables. Smyth (1981) found m(1 + x + y) = (3√3 / 4π) L(χ₋₃, 2), the first of many identities linking Mahler measures to L-values (Boyd's conjectures, Deninger's interpretation). By Lawton's theorem (1983, conjectured by Boyd), multivariable measures are limits of one-variable ones, e.g. m(P(x, xᴺ)) → m(P(x, y)). - Elliptic curves. The canonical height plays the role of log M(α)/d; the elliptic Lehmer problem is open too, with partial results by Laurent (CM curves), Masser and Hindry–Silverman. 9. Status as of 2026-10-14 - Open. We found no 2025–2026 result that proves or disproves the conjecture or lowers the record M(L). Wikipedia's article (last edited 2026-06-09), recent papers (for example arXiv 2508.08003, 2025) and the Justin Sun Prize MathDB pilot register (JSP-000025, no solution, Lean or exposition entry recorded) all treat it as open, and Sac-Épée's 2025 search found no new Salem number below 1.3. - Claimed proofs. Verger-Gaugry posted claimed proofs (arXiv 1709.03771, 2017; 1911.10590, 2019). They have not been accepted as settling the problem; specialists have reported a fatal error in the 2017 version, and current literature still calls the statement a conjecture. - Recent related work (not resolving it). A May 2026 MPIM Bonn talk proves a conjecture of Moree and Herrera-Poyatos on numerical-semigroup polynomials assuming Lehmer's conjecture. In August 2026, Alfarano, Bianucci, Lalín and Ringeling studied Boyd's multivariable Mahler measure conjectures over 250,000 cases with statistics assisted by Claude and transformer experiments (arXiv 2608.00615); this concerns L-value identities, not the lower bound. arXiv 2609.14479 (September 2026) proves a weak form of Chinburg's conjecture on multivariable measures. 10. Proved versus open, as of 2026-10-14 Proved: 1. M(P) = 1 exactly for ±xᵏ times products of cyclotomic polynomials, and only for them (Kronecker; section 3). 2. For each degree there is a smallest measure above 1 (section 2(f)). 3. Non-reciprocal P with P(0) ≠ 0 have M(P) ≥ θ₀ = 1.3247..., sharp (Smyth 1971). 4. log M(α) ≥ ¼ (log log d / log d)³ for algebraic α of degree d ≥ 2 that are not roots of unity (Dobrowolski 1979 in shape; Voutier 1996 constant). 5. Lehmer's conjecture holds for totally real α, for α in abelian extensions of ℚ, for polynomials with all coefficients odd, and for measures coming from integer symmetric matrices (with the sharp constant λ in the last case). 6. Schinzel–Zassenhaus (the house version) is proved (Dimitrov 2019). 7. No measure lies in (1, M(L)) for algebraic numbers of degree ≤ 54 (Mossinghoff–Rhin–Wu 2008 with Smyth for odd degrees); all measures below 1.3 are known up to degree 44. Open: 1. Lehmer's conjecture: is there any μ > 1 that works in every degree? 2. Is M(L) = 1.17628... the smallest measure above 1? (The "strong" form.) 3. Are Salem numbers bounded away from 1? (Implied by 1; the converse is unknown.) 4. Any general lower bound on log M(α) of a better order than (log log d / log d)³. 5. The elliptic and higher-dimensional analogues. 11. What is original here - A self-contained route from the definition to Kronecker's theorem through root squaring alone, with the termination analysis that turns the proof into an exact decision procedure for M = 1 (fixed point after at most v + 1 squarings when M = 1; a binomial bound broken once 2ᵏ log M ≥ n log 2 when M > 1). - A certified enclosure of M(P) from the same iteration, using only exact integer arithmetic until a final logarithm: after k squarings, with Q = Pₖ of degree n, max_j |Q_j| / C(n, j) ≤ M(P)^{2ᵏ} ≤ ‖Q‖₂, by 2(d) and 2(e). The interval shrinks like O(n / 2ᵏ) in log M. (lehmer.mahler_bounds.) - An exact computation of Lehmer's numbers Δₖ as circulant determinants (fraction-free Bareiss elimination). - lehmer.py and test_lehmer.py: Mahler measure from Aberth–Ehrlich roots after exact removal of cyclotomic factors, Jensen-formula cross-check, cyclotomic polynomials and factorisation, the Kronecker test, the certified bounds, Lehmer's numbers, and the two family searches of section 7, in plain Python with no dependencies. Nothing here claims progress on the conjecture itself. 12. Using the companion code python lehmer.py measure 1 1 0 -1 -1 -1 -1 -1 0 1 1 # Lehmer's polynomial, highest degree first python lehmer.py search 5 # reciprocal degree-10 {-1,0,1} family, M < 1.3 python lehmer.py smyth 6 # Smyth's bound on non-reciprocal polynomials python lehmer.py pierce 20 1 -1 -1 # Lehmer's Delta_k for x^2 - x - 1 python -m unittest test_lehmer # or: python test_lehmer.py In Python code, polynomials are lists of integer coefficients, lowest degree first. 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The Kakeya conjecture in October 2026: what is proved, what changed in 2025-2026, what is open What is proved, what changed in 2025-2026, and what is open. This is an exposition of the Kakeya set conjecture in Euclidean space and over finite fields, with complete proofs of the elementary results (the planar Córdoba-type bound, Dvir's finite-field theorem, and several small finite-field facts), a dated record of the dimension bounds, and a tested, dependency-free Python companion (kakeya.py) that checks Kakeya sets in AG(n, q), builds the two explicit constructions below, and computes every finite-field bound quoted here. New in version 1.1: the first exact finite-field Kakeya number in dimension three over a field with more than two elements. The smallest Kakeya set in F₃³ has exactly 13 points (section 7), proved by an exhaustive search that runs in seconds with no solver. The release also gives smaller Kakeya sets than the known constructions in F₃⁴, F₃⁵, F₄³ and F₄⁴, with every set published and re-checked. By MV2 of Munim, Inc. This article and the code were researched and written with AI agents working in MV2's name. Every statement below is either proved here or cited to a source; section 10 separates what is proved from what is open, and section 12 lists what is new here. Corrections are welcome by issue or pull request. Version 1.1, 2026-10-14. Text and data CC BY 4.0, code Apache-2.0. Related problem record: Justin Sun Prize JSP-000015, "Kakeya dimension conjecture (general dimension)" (MathDB #315983). --- 1. The question A Kakeya set (or Besicovitch set) in R^n is a compact set containing a unit line segment in every direction. Kakeya set conjecture. Every Kakeya set in R^n has Hausdorff dimension n (and hence also Minkowski dimension n). The Justin Sun Prize catalogue states it as: "Must a set containing a unit line segment in every direction have full Hausdorff dimension in its Euclidean ambient space?" Its title says "general dimension", so the open content in October 2026 is n ≥ 4 (section 10). History in three lines. Kakeya (1917) asked for the smallest area of a planar region in which a unit needle can be turned around. Besicovitch (1919, published again in 1928) showed that a planar set containing a unit segment in every direction can have measure zero, which answers the needle problem in a surprising way: the area can be made arbitrarily small. Since such sets can be null, the right question is how large they must be in the sense of dimension, and that is the conjecture above. It sits underneath a chain of central problems in harmonic analysis: the Kakeya maximal function conjecture, the Fourier restriction conjecture (JSP-001041 in the same pilot) and the Bochner-Riesz conjecture. Bochner-Riesz implies restriction (Tao 1999), restriction implies the Kakeya maximal conjecture, and that implies the set conjecture. Fefferman's 1971 disc multiplier theorem used Besicovitch's construction to show that the ball multiplier is unbounded on L^p(R^n) for p ≠ 2, n ≥ 2; that is where Kakeya sets entered Fourier analysis. 2. Two notions of size For a bounded set E ⊂ R^n let N(E, δ) be the least number of δ-balls covering E. - Minkowski (box) dimension: the lower and upper limits of log N(E, δ) / log(1/δ) as δ → 0. Equivalently, the δ-neighbourhood E_δ has volume about δ^(n − dim). - Hausdorff dimension: the infimum of s for which E can be covered by balls of radii r_i with Σ r_i^s arbitrarily small. It is always at most the lower Minkowski dimension, so a Hausdorff lower bound is the stronger statement. Both are computed through δ-tubes: a δ-tube is a cylinder of length 1 and radius δ around a unit segment (in the plane, a 1 × δ rectangle). If K is a Kakeya set, its δ-neighbourhood contains a δ-tube in every direction, so a lower bound on |∪ T| for δ-tubes pointing in δ-separated directions gives a lower bound on |K_δ| and hence on the Minkowski dimension. Hausdorff bounds need the same estimate when each tube is replaced by a fixed fraction of itself (Zahl's survey, Conjecture 1.2(B)). 3. The plane, with a proof Theorem (Davies 1971). Every Kakeya set in R² has Hausdorff dimension 2. Here is the complete argument for the Minkowski version, which shows the mechanism that every later proof tries to extend. Lemma 1. Two δ-tubes in R² whose directions make an angle θ ∈ (0, π/2] intersect in a set of area at most δ²/sin θ ≤ (π/2) δ²/θ. Proof. Each tube lies in an infinite strip of width δ. Two strips of width δ at angle θ intersect in a parallelogram whose sides have length δ/sin θ and whose height is δ, so its area is δ²/sin θ. Finally sin θ ≥ 2θ/π on [0, π/2]. ∎ Theorem 2. Let T_1, ..., T_N be δ-tubes in R² with directions θ_i = iδ, N = ⌊π/δ⌋. Then |∪ T_i| ≥ Nδ / (1 + π(1 + ln N)), which is about 1/log(1/δ). Proof. Let f = Σ 1_{T_i}. By Cauchy-Schwarz, (∫ f)² ≤ |supp f| · ∫ f², so |∪ T_i| ≥ (Σ |T_i|)² / Σ_{i,j} |T_i ∩ T_j|. Here Σ |T_i| = Nδ, since each tube is a 1 × δ rectangle. For fixed i and j ≠ i, put m = min(|i − j|, N − |i − j|); since Nδ ≤ π, the angle between the two tubes (taken in (0, π/2]) is at least mδ, and each value m = 1, ..., ⌊N/2⌋ occurs for at most two values of j. By Lemma 1, Σ_j |T_i ∩ T_j| ≤ δ + 2 Σ_{m ≤ N/2} (π/2)δ²/(mδ) ≤ δ(1 + π(1 + ln N)). Summing over i and dividing gives the bound. ∎ Since the δ-neighbourhood of a planar Kakeya set contains such a family (it contains a 1 × δ rectangle around each of its unit segments), |K_δ| ≳ 1/log(1/δ), and N(K, δ) ≳ δ^(−2)/log(1/δ): the Minkowski dimension is 2. The logarithm is a real loss, not an artefact: Besicovitch's sets show that |K_δ| can tend to 0. Córdoba (1977) turned this computation into the sharp L² bound for the planar Kakeya maximal function. Why the argument stops at n = 2. In R^n the same computation only controls pairs of tubes. Tubes can now meet in complicated configurations (many tubes through one point lying near a plane, or arranged along a ruled surface), and pair counting alone gives only the dimension bound (n + 1)/2 (Drury 1983; Christ, Duoandikoetxea and Rubio de Francia 1986). Every later record comes from a new way to rule out such configurations. 4. The record in R^n, dated | Year | Who | Result | | --- | --- | --- | | 1971 | Davies | dimension 2 in R² | | 1977 | Córdoba | sharp Kakeya maximal estimate in R² | | 1983, 1986 | Drury; Christ-Duoandikoetxea-Rubio de Francia | (n + 1)/2 in R^n | | 1991 | Bourgain | first improvement on (n + 1)/2 ("bush" argument) | | 1995 | Wolff | (n + 2)/2 in R^n (hairbrush; 5/2 in R³, 3 in R⁴) | | 1999 | Bourgain | (13<em>n</em> + 12)/25 via additive combinatorics ("arithmetic" method) | | 1999 | Katz-Tao | Minkowski (4<em>n</em> + 3)/7, Hausdorff (6<em>n</em> + 5)/11 | | 2000 | Katz-Łaba-Tao | upper Minkowski 5/2 + ε in R³ | | 2001 | Łaba-Tao | upper Minkowski (n + 2)/2 + ε_n for n ≥ 4 | | 2002 | Katz-Tao | Minkowski n/α + (α − 1)/α with α³ − 4α + 2 = 0 (α = 1.675...); Hausdorff (2 − √2)(n − 4) + 3 | | 2018 | Guth-Zahl, with Zahl and Katz-Rogers (polynomial Wolff axioms) | 3 + 1/40 in R⁴ | | 2019 | Katz-Zahl | Hausdorff 5/2 + ε in R³ | | 2021 | Katz-Zahl (planebrush) | 3.059 in R⁴ | | 2019-2021 | Zahl; Hickman-Rogers-Zhang | polynomial-partitioning ("broad") estimates; new bounds in some dimensions n ≥ 7 | | 2025 | Wang-Zahl | dimension 3 in R³: the conjecture holds for n = 3 | | 2026 | Rai Choudhuri (preprint, not refereed) | 13/4 − (12951 − 54√10593)/41972 = 3.0738... in R⁴ | The 2002 Katz-Tao bounds are quoted from Theorem 1.1 of their paper; the paper notes that its Hausdorff bound is better than its Minkowski bound for n ≤ 23. The R⁴ value 3.0738 is Theorem 1 of arXiv 2610.12480 (submitted 30 August 2026), which combines a slightly improved Guth-Zahl trilinear estimate, obtained by applying Wang-Zahl inside thickened neighbourhoods of varieties, with the Katz-Zahl planebrush. The author says some AI assistance was used. We give its number exactly as stated and have not checked the proof. For n ≥ 5 we do not attempt a per-dimension table: the best bounds come from a mix of the Katz-Tao arithmetic bounds and the polynomial-method estimates, and Zahl's 2025 survey (ICM 2026) does not tabulate them. All known bounds for large n grow like cn with c well below 1 (for example n/1.675 for the Minkowski dimension), so the gap to n widens with the dimension. 5. What Wang and Zahl proved, and how Theorem (Wang-Zahl 2025). Every Kakeya set in R³ has Hausdorff dimension 3 and Minkowski dimension 3. The proof is spread over three papers: the sticky case ("Sticky Kakeya sets and the sticky Kakeya conjecture", J. Amer. Math. Soc. 39 (2026) 515-585), an Assouad-dimension paper (Invent. Math. 241 (2025)), and the general case, arXiv 2502.17655 (February 2025), which reduces the general case to the sticky one by induction on scales. Guth, Wang and Zahl posted a streamlined proof in January 2026 (arXiv 2601.14411); Guth's Séminaire Bourbaki survey appeared in April 2026 (arXiv 2604.03416), and Alex Cohen circulated detailed checking notes the same month. Hong Wang received a 2026 Fields Medal at the International Congress in Philadelphia on 23 July 2026, with this work cited. As of the sources we read, the main 2025 paper is still listed as a preprint. The shape of the argument, following Guth's survey: 1. Stickiness. A family of δ-tubes is sticky if, at every coarser scale ρ, the tubes group into about ρ^(1−n) fat ρ-tubes, the minimum possible. Sticky families are the hardest case. 2. Grains. In a worst-case sticky family the union locally looks like a product of a small square with a fractal set A: the tubes stack into "grains" (Katz-Łaba-Tao). 3. The obstruction is complex conjugation. Following grains around a closed cycle forces an approximate identity A + s(x)y + s̃(y)x ≈ A. The model solution is R + x̄y + ȳx = R inside C, and it really happens: the Heisenberg group in C³ gives a tube arrangement over C that is sticky and grainy with the 5/2 behaviour. This is why every method that ignores the order structure of R stalls at 5/2. 4. Sum-product kills it over R. Bourgain's discretised sum-product theorem says R has no such approximate subrings, which rules out the conjugation structure; repeating a small gain across scales gives the sharp result. 5. Reduction to sticky. In a non-sticky family, the densest convex sets containing many tubes are handled by induction on scales (the "convex Wolff axioms" theorem, Guth's Theorem 1.2). What it does not prove: the Kakeya maximal function conjecture in R³ (Wang-Zahl say so explicitly, and Zahl's survey lists obstructions to adapting the argument; a special case for SL₂ lines was proved by Green, Harris and Ou, Analysis & PDE 19 (2026)), restriction in R³, or anything in R⁴ beyond the improved bounds of section 4. 6. The finite-field problem, with proofs Wolff (1999) proposed a model problem. A Kakeya set in F_q^n is a set K containing a full line {b + td : t ∈ F_q} for every direction d ≠ 0. Write K(n, q) for the minimum size. Wolff asked whether |K| ≥ c_n q^n. Theorem (Dvir 2008). Every Kakeya set in F_q^n has at least C(q + n − 1, n) points. In particular |K| ≥ q^n/n!. Proof. The polynomials in n variables of degree at most q − 1 form a space of dimension C(q − 1 + n, n). If |K| were smaller, a nonzero such P would vanish on K (more unknowns than linear conditions). Let d = deg P, so 1 ≤ d ≤ q − 1 (d = 0 would make P a nonzero constant vanishing on the nonempty set K), and let P_d be its top-degree homogeneous part. For a direction v, K contains a line b + tv, so the one-variable polynomial t ↦ P(b + tv) has degree at most d < q and q roots; it is identically zero. Its coefficient of t^d is P_d(v), so P_d(v) = 0 for every v ≠ 0, and P_d(0) = 0 because d ≥ 1. So P_d is a nonzero polynomial of degree less than q in each variable that vanishes on all of F_q^n, which is impossible (induct on n: a nonzero one-variable polynomial of degree < q has fewer than q roots). ∎ The proof is ten lines. Nothing like it is known in R^n, where δ-tubes are not algebraic objects, so the finite-field model turned out to be much softer than the real problem. The sharp constant came later: - Dvir, Kopparty, Saraf and Sudan (2009, method of multiplicities): |K| ≥ q^n/2^n. - Bukh and Chao (Discrete Analysis 2021): |K| ≥ q^n / (2 − 1/q)^(n−1), so the density is at least 2^(1−n) asymptotically. This is sharp: constructions with density 2^(1−n) + O(1/q) exist (Dvir-Kopparty-Saraf-Sudan, Saraf-Sudan; the parabola set below). So for fixed n the first-order behaviour is settled: K(n, q) = (2^(1−n) + o(1)) q^n as q → ∞ (through odd q by the parabola set; Kyureghyan, Müller and Wang give constructions of the same density for even q). The plane is solved exactly. Blokhuis and Mazzocca (2008) proved K(2, q) = q(q + 1)/2 + (q − 1)/2 for odd q; for even q the minimum is q(q + 1)/2 (the smallest examples are classified: Blokhuis, De Boeck, Mazzocca and Storme 2014). Dover and Mellinger found by exhaustive search every attainable size of a Kakeya set in the Desarguesian planes of order q ≤ 9. For even q, Dvir's bound C(q + 1, 2) = q(q + 1)/2 is therefore exact in the plane. Four further elementary facts, each checked by the companion code: (a) The counting bound. K(n, q) ≥ the least s with s(s − 1) ≥ q(q^n − 1). Proof. K contains (q^n − 1)/(q − 1) lines in distinct directions; two distinct lines share at most one point, so the C(q, 2) pairs of points on each line are disjoint sets of pairs inside the C(s, 2) pairs of K. ∎ It is weak for large q but exact for q = 2 and n ≤ 4. (b) Products. If A ⊂ F_q^n and B ⊂ F_q^m are Kakeya sets, so is A × B, hence K(n + m, q) ≤ K(n, q) K(m, q). Proof. For a direction (d, e) with d, e ≠ 0, take lines a + td ⊂ A and b + te ⊂ B; then (a + td, b + te) ⊂ A × B. If e = 0 use (a + td, b) for any b ∈ B, and symmetrically if d = 0. ∎ (c) Over F₂ a Kakeya set is a difference basis. A line in F₂^n is a pair {x, x + d}, so K is Kakeya exactly when every nonzero vector is a sum a + b with a, b ∈ K: K is a difference basis of the group C₂^n. Banakh and Gavrylkiv computed the difference sizes of all abelian groups of order below 96, which gives | n | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | | K(n, 2) | 2 | 3 | 5 | 6 | 10 | 14 | | counting bound (a) | 2 | 3 | 5 | 6 | 9 | 12 | (Acta Sci. Math. (Szeged) 85 (2019), Tables 1 and 3, where the group is written (C₂)^n). For n = 4 the optimum is a perfect Sidon set: {0, e₁, e₂, e₃, e₄, e₁ + e₂ + e₃ + e₄} has 15 pairwise sums, and they are exactly the 15 nonzero vectors. The companion code reproves n ≤ 4 by exhaustion. Their Proposition 7.3 gives, for n ≥ 2, the two-sided bound √2 < K(n, 2)/2^(n/2) < 3/√2 (and < 2 for even n). (d) Two explicit constructions. - The parabola set (q odd). Write points as (x, t) with x ∈ F^(n−1). For each b ∈ F^(n−1) take the line {((tb_i − b_i²)_i, t) : t ∈ F}, of direction (b, 1), and add the hyperplane t = 0 for the directions (b, 0). For t ≠ 0 the slice at height t is {x : t²/4 − x_i is a square for every i}, because t²/4 − (tb − b²) = (b − t/2)². So the set has (q − 1)((q + 1)/2)^(n−1) + q^(n−1) points, density 2^(1−n) + O(1/q), and for n = 2 it has (q² + 2<em>q</em> − 1)/2 points: exactly the Blokhuis-Mazzocca minimum. - The Smyth set. K = (F^*)^n ∪ {0, 1}^n, with (q − 1)^n + 2^n − 1 points. This size was reported by Clifford Smyth in a 2008 comment on Tao's blog; here is a proof that this set works. Given a direction d, let Z be the set of coordinates where d is zero (not all of them) and 1_Z the 0/1 vector supported on Z. On the line 1_Z + td the point t = 0 is 1_Z ∈ {0, 1}^n, and for t ≠ 0 every coordinate is nonzero: 1 on Z, td_i ≠ 0 off Z. ∎ For q = 3 it gives K(n, 3) ≤ 2^(n+1) − 1, which beats the parabola set's 2^n + 3^(n−1) for every n ≥ 3, and no 0/1 vector other than (1, ..., 1) can be dropped from it (checked by the tests for n = 3). 7. Small cases: what is known exactly | (n, q) | points | best lower bound (which) | best verified upper bound (which) | | --- | --- | --- | --- | | (2, q) | q² | exact: Blokhuis-Mazzocca | exact | | (n, 2), n ≤ 6 | 2^n | exact: Banakh-Gavrylkiv | exact | | (3, 3) | 27 | 13 (new, this repository) | 13 (new); Smyth 15 | | (4, 3) | 81 | 18 (Bukh-Chao) | 27 (new); Smyth 31 | | (5, 3) | 243 | 32 (Bukh-Chao) | 56 (new); Smyth 63 | | (3, 4) | 64 | 21 (Bukh-Chao) | 28 (new); Smyth 34 | | (4, 4) | 256 | 48 (Bukh-Chao) | 64 (new); Smyth 96 | | (3, 5) | 125 | 39 (Bukh-Chao) | 53 (AlphaEvolve formula; found again here); parabola 61 | | (3, 7) | 343 | 100 (Bukh-Chao) | 128 (found here); parabola 145 | Every number in the table is recomputed by python kakeya.py bounds N Q, every upper bound is a set that the tests build or load and check line by line, and the lower bound 13 is re-proved by the tests (below). The 2025 AlphaEvolve paper (Georgiev, Gómez-Serrano, Tao and Wagner, arXiv 2511.02864) gives constructions in F_p³ for primes p ≡ 1 (mod 4) of size p³/4 + 7<em>p</em>²/8 − 1/8, which is 53 at p = 5; our solver found a 53-point set in F₅³ independently, and did not beat it in 25 minutes. Theorem (new). K(3, 3) = 13: the smallest Kakeya set in F₃³ has exactly 13 points. Upper bound. The set (0,0,0) (1,0,0) (2,0,0) (1,1,0) (0,0,1) (2,0,1) (0,1,1) (1,1,1) (0,2,1) (2,2,1) (0,0,2) (2,1,2) (0,2,2) contains a line in each of the 13 directions of F₃³ (over F₃ a line is a 3-term progression b, b + d, b + 2<em>d</em>): | direction | line in the set | direction | line in the set | | --- | --- | --- | --- | | (1,0,0) | (0,0,0) (1,0,0) (2,0,0) | (1,1,0) | (0,0,1) (1,1,1) (2,2,1) | | (0,1,0) | (0,0,1) (0,1,1) (0,2,1) | (1,2,0) | (0,2,1) (1,1,1) (2,0,1) | | (0,0,1) | (0,0,0) (0,0,1) (0,0,2) | (1,1,1) | (0,0,2) (1,1,0) (2,2,1) | | (0,1,1) | (0,0,0) (0,1,1) (0,2,2) | (1,1,2) | (0,2,1) (1,0,0) (2,1,2) | | (0,1,2) | (2,0,0) (2,1,2) (2,2,1) | (1,2,1) | (0,2,2) (1,1,0) (2,0,1) | | (1,0,1) | (0,0,2) (1,0,0) (2,0,1) | (1,2,2) | (0,2,2) (1,1,1) (2,0,0) | | (1,0,2) | (0,1,1) (1,1,0) (2,1,2) | | | Lower bound. Suppose K is a Kakeya set with at most 12 points. Translating K so that its line in direction e₁ becomes the axis {(0,0,0), (1,0,0), (2,0,0)} keeps it a Kakeya set of the same size, and adding points keeps it Kakeya, so there would be a 12-point Kakeya set containing the axis. There are C(24, 9) = 1,307,504 such 12-point sets, and none of them contains a line in all 13 directions. kakeya_sets_through_axis checks every one with bitmasks, with no solver, in a few seconds; test_k33_is_thirteen runs it on every test run. Separately, two independent integer-programming solvers (OR-Tools CP-SAT, with and without symmetry fixing, and HiGHS through SciPy) each proved 13 optimal in MV2's run kakeya-min-v1. ∎ So the sequence K(n, 3) begins 3, 7, 13. The minimum set is not a "lift" of a planar one in the obvious way: its slices by the last coordinate have 4, 6 and 3 points, and none of them is a planar Kakeya set (that needs 7 points). Why this is new. Lund, Saraf and Wolf (arXiv 1609.01048) wrote that no exact value is known in any dimension other than 2. The q = 2 values follow from Banakh and Gavrylkiv's difference bases (section 6(c)); for q ≥ 3 and n ≥ 3 we found no exact value in the papers cited here, in the AlphaEvolve paper (which treats asymptotic constructions), on OEIS, or in answer to the 2008 question on Tao's blog about the smallest subset of F₃^n with a 3-term progression in every direction. If an earlier computation exists, we would like to cite it. New upper bounds. The same run found Kakeya sets of 27 points in F₃⁴, 56 in F₃⁵, 28 in F₄³ and 64 in F₄⁴. Each is smaller than every explicit construction quoted in this article: the Smyth set (31, 63, 34, 96), the parabola set (43 and 113 for q = 3), products of smaller optimal sets (for example 3 · 13 = 39 and 7 · 7 = 49 in F₃⁴, 10 · 10 = 100 in F₄⁴), and the Kyureghyan-Müller-Wang construction for q = 4 (46 in F₄³ by their formula (2)). They are not claimed optimal: the solvers stopped at their time limits. CP-SAT reported a lower bound of 24 for F₄³; we do not use it, because it was not certified independently. The 27-point set in F₃⁴ has slices of 13, 6 and 8 points by its last coordinate, which suggests a construction lifting F₃³ sets to F₃⁴; MV2 is testing that next. All sets, solver logs and timings are in data/kakeya-min-v1/ (results.json: for every case the runs of each solver and a certificate, the smallest set found as a list of coordinates; results.txt: one line per case). solve.py reproduces the run (pip install -r requirements-solve.txt, then python solve.py --out-dir out; about 3.3 hours on 4 cores). Open, as far as we can tell: every other exact value with n ≥ 3 and q ≥ 3. The smallest open cases are K(4, 3) ∈ [18, 27] and K(3, 4) ∈ [21, 28]. 8. Variants that are open or recently settled - Z/NZ and p-adic rings. Over the rings Z/*N*Z the analogue of Dvir's theorem was proved by Dhar (Advances in Combinatorics 2024, general N), and Arsovski proved the p-adic Kakeya conjecture (J. Amer. Math. Soc. 37 (2024)). - Nikodym and Furstenberg sets. The Furstenberg set conjecture in the plane was settled by Ren and Wang (2024), building on Orponen-Shmerkin; Guth cites it as a hint that sticky cases are key. - Plany sets in F_q⁴. Łaba, Rai Choudhuri and Zahl (Res. Math. Sci. 13 (2026)) give a finite-field exposition of the Katz-Zahl planebrush. 9. Claimed proofs Short claimed proofs of the general conjecture appear from time to time. A claim deserves attention only if it handles the Heisenberg obstruction of section 5 (any argument that works verbatim over C proves something false) and recovers the Wang-Zahl theorem as a special case. We do not list individual unrefereed claims. 10. Proved versus open, as of 2026-10-14 Proved. - n = 2: every Kakeya set has dimension 2 (Davies 1971); the maximal conjecture holds (Córdoba 1977). - n = 3: every Kakeya set has Hausdorff and Minkowski dimension 3 (Wang-Zahl 2025; three papers, one published in JAMS 2026, the main one still a preprint; a streamlined proof exists). - All n: Hausdorff dimension ≥ max((n + 2)/2, (2 − √2)(n − 4) + 3) and the Katz-Tao Minkowski bound; in R⁴, at least 3.059 (Katz-Zahl, published) and 3.0738 (2026 preprint). - Finite fields: |K| ≥ q^n/(2 − 1/q)^(n−1), sharp to first order; the plane exactly; q = 2 exactly for n ≤ 6; K(3, 3) = 13 (new here, by exhaustive computation that anyone can rerun). Open. - The Kakeya set conjecture for every n ≥ 4 (this is the open content of JSP-000015). - The Kakeya maximal function conjecture for every n ≥ 3. - The restriction and Bochner-Riesz conjectures for n ≥ 3. - Exact finite-field values K(n, q) for n ≥ 3, q ≥ 3 other than K(3, 3); the second-order term in K(n, q). 11. Why this matters beyond the problem Kakeya estimates control how much wave packets pointing in different directions can overlap, which is the core difficulty in restriction theory, Bochner-Riesz summation, local smoothing for the wave equation and decoupling. Dvir's finite-field theorem became a standard tool in theoretical computer science (randomness mergers and extractors) and set off the modern wave of the polynomial method in combinatorics, which went on to the joints theorem and the cap-set bound. 12. What is original here - The exact value K(3, 3) = 13 (section 7), with a solver-free proof that reruns in seconds, an explicit 13-point set and a table of its 13 lines. As far as we found, it is the first exact finite-field Kakeya number in dimension at least 3 over a field with more than two elements. - Smaller Kakeya sets in F₃⁴ (27), F₃⁵ (56), F₄³ (28) and F₄⁴ (64) than the constructions we found in the literature, all published as coordinate lists and re-checked. - A dated, sourced record of the dimension bounds through October 2026, including the August 2026 R⁴ preprint and the 2026 publication and recognition of the Wang-Zahl work. - Self-contained proofs of the planar bound, Dvir's theorem, the counting bound, the product lemma, the parabola construction and its exact size, and the Smyth construction (whose proof we did not find written out). - The identification K(n, 2) = Δ[C₂^n] with difference bases, which imports exact values for n ≤ 6 from group theory. We did not find this identification stated in the Kakeya literature we read; it is elementary. - The small-case table of section 7 with every bound recomputed and every upper bound verified by machine, and a precise statement of which exact values are open. 13. Using the companion code kakeya.py needs only Python 3.8+ and no packages. python kakeya.py bounds 3 3 # counting, Dvir, Bukh-Chao, parabola and Smyth sizes python kakeya.py smyth 3 3 # print the 15-point Kakeya set in F_3^3 python kakeya.py parabola 3 5 # print the 61-point Kakeya set in F_5^3 python kakeya.py check 3 3 set.txt # check a set: one point per line, coordinates separated by spaces python kakeya.py brute 2 5 # exact minimum by brute force (small planes only) python kakeya.py exhaust 3 3 12 # no Kakeya set of 12 points in F_3^3 (a few seconds) python kakeya.py exhaust 3 3 13 # print a 13-point one python kakeya.py records # the exact values and bounds of section 7 python -m unittest -v test_kakeya Fields supported: F_p for primes p, and F₄, F₈, F₉, F₁₆. 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