Point sets avoiding near-integer distances

Abstract

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