Unit and distinct distances in typical norms

Abstract: Erd\H{o}s' unit distance problem and Erd\H{o}s' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of distinct distances, respectively, determined by $n$ points in the Euclidean plane. The question of what happens in these problems if one considers normed spaces other than the Euclidean plane has been raised in the 1980s by Ulam and Erd\H{o}s and attracted a lot of attention over the years. We give an essentially tight answer to both questions for almost all norms on $\mathbb{R}^d$, in a certain Baire categoric sense. For the unit distance problem we prove that for almost all norms $|.|$ on $\mathbb{R}^d$, any set of $n$ points defines at most $\frac{1}{2} d \cdot n \log_2 n$ unit distances according to $|.|$. We also show that this is essentially tight, by proving that for every norm $|.|$ on $\mathbb{R}^d$, for any large $n$, we can find $n$ points defining at least $\frac{1}{2}(d-1-o(1))\cdot n \log_2 n$ unit distances according to $|.|$. For the distinct distances problem, we prove that for almost all norms $|.|$ on $\mathbb{R}^d$ any set of $n$ points defines at least $(1-o(1))n$ distinct distances according to $|.|$. This is clearly tight up to the $o(1)$ term. We also answer the famous Hadwiger--Nelson problem for almost all norms on $\mathbb{R}^2$, showing that their unit distance graph has chromatic number $4$. Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matou\v{s}ek, Brass-Moser-Pach, Chilakamarri, and Robertson. The proofs combine combinatorial and geometric ideas with tools from Linear Algebra, Topology and Algebraic Geometry.