Better bounds for planar sets avoiding unit distances

TL;DR

This paper establishes new upper bounds on the maximum density of planar sets avoiding unit distances, proving a conjecture for structured sets and improving the general bound using harmonic analysis.

Contribution

It proves that structured 1-avoiding sets in high dimensions have density less than 1/2^n and refines the upper bound for the planar case using advanced mathematical techniques.

Findings

Structured 1-avoiding sets have density less than 1/2^n.

Confirmed Erdős's conjecture for sets with block structure in the plane.

Upper bound for m_1(ℝ^2) is now 0.258795.

Abstract

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.