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4-slope Kakeya-type sum-difference constant (C3cC_{3c})

4-slope Kakeya-type sum-difference constant: determine C3cC_{3c} or improve its bounds; the repository records 1.67473389 (1.6747338950414058*) (lower) and 1.75 (upper).

MSC 11B13 · Additive bases, including sumsets

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

C3c=SD({0,1,2,∞};−1)C_{3c} = SD(\{0,1,2,\infty\};-1) is the least exponent such that one has the inequality

∣A−GB∣≤max⁡(∣A∣,∣B∣,∣A+GB∣,∣A+G2B∣)C3c|A \stackrel{G}{-} B| \leq \max(|A|, |B|, |A \stackrel{G}{+} B|, |A \stackrel{G}{+} 2B|)^{C_{3c}}

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

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

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Background

  • Has many other formulations [GR2019], including an entropy formulation: C3cC_{3c} is the smallest constant such that for any pair of discrete random variables X,YX,Y one has
H(X−Y)≤C3cmax⁡(H(X),H(Y),H(X+Y),H(X+2Y)).H(X-Y) \leq C_{3c} \max( H(X), H(Y), H(X+Y), H(X+2Y)).

This entropy formulation has been used to attain all known lower bounds.

  • Related to the arithmetic Kakeya conjecture [KT2002], [GR2019], which considers other sets of slopes than 0,1,2,∞0,1,2,\infty.

References (from the repository page)

  • [A2026] Astor, T. Improved Arithmetic Kakeya-Type Counterexamples. TBA (2026)
  • [G2026] Griego, Sebastian. 26-point entropy certificate for C3cC_{3c}, submitted to this repository (2026).
  • [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.
  • [L2026] Lin, Yongxi. 147-point entropy certificate for C3cC_{3c}, certificate and verifiers (pinned revision 62621d2), submitted to this repository (2026).
  • [MI2026] Mosaic Intelligence (@111111). 95-point entropy certificate for C3cC_{3c}, 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 1.67473389 (1.6747338950414058*); best known upper bound 1.75. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
2 Trivial
2−14=1.752 - \frac{1}{4} = 1.75 [KT1999]

Known lower bounds

Bound Reference Comments
1.612261.61226 [L2015]
1.6681.668 [GGSWT2025]
1.674711.67471 [A2026]
1.674733891.67473389 [G2026] Entropy construction on a 26-point support.
1.67473389502082491.6747338950208249 [MI2026] Entropy construction on a 95-point support.
1.67473389504140581.6747338950414058 [L2026] Entropy construction on a 147-point support, exact rational weights at denominator 1032010^{320}. Certified C3c≥1.674733895041405870063135756722213999136383713818148696811828C_{3c} \geq 1.674733895041405870063135756722213999136383713818148696811828, with H(X−Y)=1.90118266043426947934…H(X-Y) = 1.90118266043426947934\ldots and H(X)=H(Y)=H(X+Y)=H(X+2Y)=1.13521477415805507651…H(X) = H(Y) = H(X+Y) = H(X+2Y) = 1.13521477415805507651\ldots (natural logarithm), so no constraint is slack. Improves [MI2026] by 2.06×10−112.06 \times 10^{-11}.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — 4-slope Kakeya-type sum-difference constant ($C_{3c}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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