Point sets avoiding near-integer distances

Abstract

Let $d\in \mathbb{N}$, $\hat{I}´\in (0,1/2)$, and $X>0$. Denote by $N_d(X,\hat{I}´)$ the maximum number of points in a subset of the closed Euclidean ball of radius $X$ in ${\mathbb{R}}^d$ such that every pairwise distance is at least $\hat{I}´$ away from any integer. In the planar case, Sárközy proved that for every $\epsilon >0$, $N_2(X,\hat{I}´)=\hat{I}{\text{c◯}}_{\hat{I}}´(X^{1/2-\epsilon })$ as $X\to \infty$ whenever $\hat{I}´$ is sufficiently small in terms of $\epsilon$, while Konyagin proved the almost matching upper bound $N_2(X,\hat{I}´)=O_{\hat{I}}´(X^{1/2})$. We study this problem in higher dimensions, addressing a question of Erdős and Sárközy. Extending Sárközy's construction, we show that for every $\epsilon >0$, $N_3(X,\hat{I}´)=\hat{I}{\text{c◯}}_{\hat{I}}´(X^{1-\epsilon })$ for $\hat{I}´$ sufficiently small in terms of $\epsilon$. We also provide a lifting lemma from integer distance sets to sets avoiding near-integer distances via bilipschitz embeddings of snowflaked Euclidean spaces. This allows us to prove a linear lower bound $N_4(X,\hat{I}´)=\hat{I}{\text{c◯}}_{\hat{I}}´(X)$ for all sufficiently small $\hat{I}´$. Finally, adapting Konyagin's approach, we prove the upper bound $N_d(X,\hat{I}´)=O_{d,\hat{I}´}(X^{d/2})$ for all $d\in \mathbb{N}$.

15 pages, 1 figure