Every n points in the plane contain (m1 + o(1)) n points with no unit distance

Abstract

Let $f(n)$ be the largest integer such that every set of $n$ points in the plane contains $f(n)$ points no two at distance $1$, and let $m_1$ be the supremum of the upper densities of Lebesgue measurable planar sets with no two points at distance $1$. Larman and Rogers observed that $f(n) \ge m_1 n$. We prove that $f(n)/n \to m_1$; that is, $f(n) = (m_1 + o(1))n$. This is the first half of a conjecture of Dúcz and Varga, and with the bound $m_1 \le 0.247$ of Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki it gives $f(n) \le (0.247 + o(1))n$. The proof builds a random independent subset of the algebraic plane, invariant under algebraic translations and rotations, whose probability of containing the origin is at least the worst finite independence ratio, and turns it into a measurable planar set of at least that upper density. Its only non-classical input is the Haar rigidity theorem for wild character laws from the OpenAI preprint The Euclidean plane is not five-colorable, which has a machine-checked Lean proof. The whole argument has been formalized in Lean 4 on top of it.