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
\(A \stackrel{G}{\pm} rB := \{ a \pm rb: (a,b) \in G\}.\)
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. |
| $1.6747338950414058$ |
[L2026] |
Entropy construction on a 147-point support, exact rational weights at denominator $10^{320}$. Certified $C_{3c} \geq 1.674733895041405870063135756722213999136383713818148696811828$, with $H(X-Y) = 1.90118266043426947934\ldots$ and $H(X) = H(Y) = H(X+Y) = H(X+2Y) = 1.13521477415805507651\ldots$ (natural logarithm), so no constraint is slack. Improves [MI2026] by $2.06 \times 10^{-11}$. Independently replayed [T2026]: the ratio of the certificate’s law (sha256 b3dc9587ba7a…) is enclosed as $1.67473389504140587006\ldots$, and the 60-digit bound above holds. |
| $1.674894211307116$ |
[J2026] |
Entropy construction from $n = 10^{20}$ independent base-55 digits: 39041372760532870645 drawn from a law on 51 points of $\mathbb{Z}^2$ and 60958627239467129355 from a law on 47 points (exact rational weights, multiples of $10^{-50}$), with $x-y$ injective on each support and all five forms at most 27 in absolute value. Per digit (natural logarithm), $H(X-Y) = 1.90144092556070111045748\ldots$ and $\max(H(X),H(Y),H(X+Y),H(X+2Y)) = H(X+2Y) = 1.13526031239715368784461\ldots$; the four agree to about 21 digits. Certified $C_{3c} \geq 1.674894211307116$. Improves [L2026] by $1.603 \times 10^{-4}$. Independently replayed [T2026b]: the certificate (sha256 8b6612ef9134…) gives a ratio enclosed as $1.67489421130711622876434373706\ldots$ in exact rational and interval arithmetic. |
- 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.
- [J2026] Jaswal, Manvir. Two-law entropy certificate for $C_{3c}$, certificate and checks, submitted to this repository (2026).
- [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 $C_{3c}$, certificate and verifiers (pinned revision
62621d2), submitted to this repository (2026).
- [MI2026] Mosaic Intelligence (@111111). 95-point entropy certificate for $C_{3c}$, certificate archive, submitted to this repository (2026).
- [T2026] Toledo, Carlos. Independent replay of the 147-point [L2026] certificate for $C_{3c}$: exact rational pushforwards and outward-rounded logarithms, in two implementations that share no code with each other or with the certificate’s checkers. Report; code and record at commit
422b952, archived as doi:10.5281/zenodo.23171167 (2026).
- [T2026b] Toledo, Carlos. Independent replay of the two-law [J2026] certificate for $C_{3c}$: exact rational pushforwards per law, outward-rounded entropies, and the injectivity of the base-55 digit map checked (every form at most 27 in absolute value on both supports), in two implementations that share no code with each other or with the certificate’s checkers. Report (§3); code and record at commit
9925150 (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
- Used formatting of 3b.md
- The [L2026] row was prepared with AI assistance; the certificate is machine-checkable by the two independent verifier scripts linked above.
- The [J2026] row was prepared with an AI system; the certificate is machine-checkable by the two independent scripts linked above.
- The [T2026b] replay was prepared with AI assistance (Claude); its numbers are read from the record at the commit cited.
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