Better bounds for planar sets avoiding unit distances

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