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
| Bound |
Reference |
Comments |
| 2 |
Trivial |
|
| $2 - \frac{1}{4} = 1.75$ |
[KT1999] |
|
Known lower bounds
| Bound |
Reference |
Comments |
| $1.61226$ |
[L2015] |
|
| $1.668$ |
[GGSWT2025] |
|
| $1.67471$ |
[A2026] |
|
| $1.67473389$ |
[G2026] |
Entropy construction on a 26-point support. |
| $1.6747338950208249$ |
[MI2026] |
Entropy construction on a 95-point support. |
- Has many other formulations [GR2019], including an entropy formulation: $C_{3c}$ is the smallest constant such that for any pair of discrete random variables $X,Y$ one has
\(H(X-Y) \leq C_{3c} \max( H(X), H(Y), H(X+Y), H(X+2Y)).\) 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 $0,1,2,\infty$.
References
- [A2026] Astor, T. Improved Arithmetic Kakeya-Type Counterexamples. TBA (2026)
- [G2026] Griego, Sebastian. 26-point entropy certificate for $C_{3c}$, 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.
- [MI2026] Mosaic Intelligence (@111111). 95-point entropy certificate for $C_{3c}$, 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.
Contribution notes