Lisa Sauermann: Unit distances and distinct distances in typical norms

Given n points in the plane, how many pairs among these points can have distance exactly 1? More formally, what is the maximum possible number of unit distances among a set of n points in the plane? This problem is a very famous and still largely open problem, called the Erdös unit distance problem. One can also study this problem for other norms on R² (or more generally on R^d for any dimension d) that are different from the Euclidean norm. This direction has been suggested in the 1980s by Ulam and Erdös and attracted a lot of attention over the years. We give an almost tight answer to this question for almost all norms on R^d (for any given d). Furthermore, for almost all norms on R^d, we prove an asymptotically tight bound for a related problem, the so-called Erdös distinct distances problem. Joint work with Noga Alon and Matija Bucić.