4-slope Kakeya-type sum-difference constant

Description of constant

$C_{3c} = SD({0,1,2,\infty};-1)$ is the least exponent such that one has the inequality \(|A \stackrel{G}{-} B| \leq \max(|A|, |B|, |A \stackrel{G}{+} B|, |A \stackrel{G}{+} 2B|)^{C_{3c}}\) whenever $A, B$ are finite subsets of reals and $G \subset A \times B$, where \(A \stackrel{G}{\pm} rB := \{ a \pm rb: a \in A, b \in B\}.\)

Known upper bounds

Known lower bounds

Additional comments and links

• Has many other formulations [GR2019], including an entropy formulation: $C_{3b}$ is the smallest constant such that for any pair of discrete random variables $X,Y$ one has \(H(X-Y) \leq C_{3b} \max( H(X), H(Y), H(X+Y), H(X+2Y)).\) 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 $0,1,2,\infty$.

References

• [A2026] Astor, T. Improved Arithmetic Kakeya-Type Counterexamples. TBA (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.
• [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.

• Contribution notes

• Used formatting of 3b.md