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arXiv · 1501.00168

Better bounds for planar sets avoiding unit distances

Tamás Keleti·Máté Matolcsi·Fernando Mário de Oliveira Filho·Imre Z. Ruzsa
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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 $m_1(\mathbb{R}^2) \leq 0.258795$.

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Tamás Keleti, Máté Matolcsi, Fernando Mário de Oliveira Filho, Imre Z. Ruzsa. 2015-10-26. Better bounds for planar sets avoiding unit distances. https://doi.org/10.1007/s00454-015-9751-5

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