An improved constant factor for the unit distance problem

Péter Ágoston, Dömötör Pálvölgyi

Published 2020-06-11Version 1

We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. Our main method is a crossing lemma for multigraphs with a better constant, which is of independent interest, as our proof is simpler than earlier proofs.