Kakeya-type sum-difference constant

Description of constant

$C_{3b} = SD({0,1,\infty};-1)$ is the least exponent such that one has the inequality \(|A \stackrel{G}{+} B| \leq \max(|A|, |B|, |A \stackrel{G}{-} B|)^{C_{13}}\) whenever $A, B$ are finite subsets of reals and $G \subset A \times B$, where \(A \stackrel{G}{\pm} B := \{ a \pm b: a \in A, b \in B\}.\)

Known upper bounds

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$2 - \frac{1}{14} = 1.92857\dots$ Wolff (unpublished)  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

Known lower bounds

Bound Reference Comments
$\frac{\log 3}{\log 2} = 1.58496\dots$ Trivial  
$\frac{\log 27}{\log (27/4)} = 1.72598\dots$ Ruzsa (unpublished)  
$1.77898$ [L2015]  
$>1.77898$ [GGSWT2025] Improved [L2015] in the eighth decimal place (AlphaEvolve)

References