A new upper bound for the size of $s$-distance sets in boxes

Abstract: Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of (non-zero) distances among points of $\mbox{$\cal G$}$: $$ d(\mbox{$\cal G$}):={d( p_1, p_2):~ p_1, p_2\in \mbox{$\cal G$}, p_1\ne p_2}. $$ Our main result is a new upper bound for the size of $s$-distance sets in boxes. More concretely, let $A_i\subseteq \mathbb R$, $|A_i|=q\geq 2$ be subsets for each $1\leq i\leq n$. Consider the box $\mbox{$\cal B$}:=\prod_{i=1}^n A_i\subseteq {\mathbb R}^n$. Suppose that $\mbox{$\cal G$}\subseteq \mbox{$\cal B$}$ is a set such that $|d(\mbox{$\cal G$})|\leq s$. Let $d:=\frac{n(q-1)}{s}$. Then $$|\mbox{$\cal G$}|\leq 2(qJ(q,d))^n.$$ We use Tao's slice rank bounding method in our proof.