Better Bounds for Planar Sets Avoiding Unit Distances

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 Erdős asserting that $m_1(\mathbb {R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb {R}^2) \le 0.258795$.