# Sets in $\mathbb{R}^d$ with slow-decaying density that avoid an unbounded collection of distances

## TL;DR

This paper constructs sets in Euclidean space with slowly decaying density that avoid an unbounded sequence of distances, demonstrating sharpness of known distance-avoidance results and generalizing to various norms.

## Contribution

It provides a novel construction of sets in $\, ext{R}^d$ with specific distance-avoidance properties and slow density decay, extending previous results to general norms.

## Key findings

- Constructed sets avoid an unbounded sequence of distances.
- Sets have density decay controlled by an arbitrary function $f(R)$.
- Generalizes to any metric induced by a norm on $\, ext{R}^d$.

## Abstract

For any $d\in \mathbb{N}$ and any function $f:(0,\infty)\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A \subseteq \mathbb{R}^d$ and a sequence $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x,y\in A$ and $\mu(A\cap B_{R_n})\geq f(R_n)\mu(B_{R_n})$ for all $n\in \mathbb{N}$, where $B_R$ is the ball of radius $R$ centered at the origin and $\mu$ is Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on $\mathbb{R}^d$.