Point sets avoiding near-integer distances

TL;DR

This paper investigates the maximum size of point sets in Euclidean spaces that avoid near-integer distances, extending known planar results to higher dimensions and establishing new bounds.

Contribution

It extends constructions to higher dimensions, introduces a lifting lemma for near-integer distance sets, and provides both lower and upper bounds in various dimensions.

Findings

In 3D, point sets can be almost linearly large in size.

A lifting lemma connects integer distance sets to near-integer distance sets.

Upper bounds scale as the square root of the volume in higher dimensions.

Abstract

Let $d \in \mathbb{N}$, $\delta \in (0, 1/2)$, and $X > 0$. Denote by $N_d(X, \delta)$ 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 $\delta$ away from any integer. In the planar case, S\'ark\"ozy proved that for every $\varepsilon > 0$, $N_2(X, \delta) = \Omega_\delta(X^{1/2-\varepsilon})$ as $X \rightarrow \infty$ whenever $\delta$ is sufficiently small in terms of $\varepsilon$, while Konyagin proved the almost matching upper bound $N_2(X,\delta) = O_\delta(X^{1/2})$. We study this problem in higher dimensions, addressing a question of Erd\H{o}s and S\'ark\"ozy. Extending S\'ark\"ozy's construction, we show that for every $\varepsilon > 0$, $N_3(X, \delta) = \Omega_\delta(X^{1-\varepsilon})$ for $\delta$ sufficiently small in terms of $\varepsilon$. 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 $N_4(X,\delta) = \Omega_\delta(X)$ for all sufficiently small $\delta$. Finally, adapting Konyagin's approach, we prove the upper bound $N_d(X, \delta) = O_{d, \delta}(X^{d/2})$ for all $d \in \mathbb{N}$.