OpenFrontiers
Join
← Back to the index
OP-11995OpenCombinatorics

Kakeya sums-differences constant (C3bC_{3b})

Kakeya sums-differences constant: determine C3bC_{3b} or improve its bounds; the repository records 1.77898 (1.77898884*) (lower) and 1.83333 (upper).

MSC 11B13 · Additive bases, including sumsetsMSC 28A75 · Length, area, volume, other geometric measure theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C3b=SD({0,1,∞};−1)C_{3b} = SD(\{0,1,\infty\};-1) is the least exponent such that one has the inequality

∣A−GB∣≤max⁡(∣A∣,∣B∣,∣A+GB∣)C3b|A \stackrel{G}{-} B| \leq \max(|A|, |B|, |A \stackrel{G}{+} B|)^{C_{3b}}

whenever A,BA, B are finite subsets of reals and G⊂A×BG \subset A \times B, where

A±GB:={a±b:a∈A,b∈B}.A \stackrel{G}{\pm} B := \{ a \pm b: a \in A, b \in B\}.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

  • Has many other formulations [GR2019], including an entropy formulation: C3bC_{3b} is the smallest constant such that for any pair of discrete random variables X,YX,Y one has
H(X−Y)≤C3bmax⁡(H(X),H(Y),H(X+Y)).H(X-Y) \leq C_{3b} \max( H(X), H(Y), H(X+Y)).
  • In [B1999] it was observed that Kakeya sets in dimension dd have Minkowski and Hausdorff dimension at least d−1C3b+1\frac{d-1}{C_{3b}} + 1. (This is no longer the best bound in any dimension.)
  • Related to the arithmetic Kakeya conjecture [KT2002], [GR2019], which considers other sets of slopes than 0,1,∞0,1,\infty.

References (from the repository page)

  • [B1999] Bourgain, J. On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal. 9 (1999), no. 2, 256-282. DOI: 10.1007/s000390050087.
  • [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
  • [GR2019] Green, B.; Ruzsa, I. Z. On the arithmetic Kakeya conjecture of Katz and Tao. Periodica Mathematica Hungarica, Volume 78, Issue 1, pp 135–151 (2019). DOI: 10.1007/s10958-018-2003-3.
  • [L2015] Lemm, Marius. New counterexamples for sums-differences. Proceedings of the American Mathematical Society, Vol. 143, No. 9 (SEPTEMBER 2015), pp. 3863-3868 (6 pages). DOI: 10.1090/proc/12731.
  • [MI2026] Mosaic Intelligence (@111111). 13-point entropy certificate for C3bC_{3b}, certificate archive, submitted to this repository (2026).
  • [KT1999] Katz, Nets Hawk; Tao, Terence. Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett. 6 (1999), no. 5-6, 625-630. DOI: 10.4310/MRL.1999.v6.n6.a3.
  • [KT2002] Katz, N. H.; Tao, T. New bounds for Kakeya problems. J. Anal. Math. 87 (2002), 231–263. DOI: 10.1007/BF02792310.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.77898 (1.77898884*); best known upper bound 1.83333. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
2 Trivial
2−114=1.92857…2 - \frac{1}{14} = 1.92857\dots Wolff (unpublished)
2−113=1.92307…2 - \frac{1}{13} = 1.92307\dots [B1999]
2−16=1.83333…2 - \frac{1}{6} = 1.83333\dots [KT1999]

Known lower bounds

Bound Reference Comments
log⁡3log⁡2=1.58496…\frac{\log 3}{\log 2} = 1.58496\dots Trivial
log⁡27log⁡(27/4)=1.72598…\frac{\log 27}{\log (27/4)} = 1.72598\dots Ruzsa (unpublished)
1.778981.77898 [L2015]
>1.77898>1.77898 [GGSWT2025] Improved [L2015] in the eighth decimal place (AlphaEvolve)
1.778988841.77898884* [MI2026] Entropy construction on a 13-point support. Asterisked: the certificate is a self-archived Zenodo deposit and the bound has not been independently replayed.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Kakeya sums-differences constant ($C_{3b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.