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 $m_1({\mathbb{R}}^2)\le 0.258795$.

16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the script

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