An incomplete attack on the upper bound of the unit distance problem

Abstract

This is an incomplete attempt to show that the upper bound of $\lesssim n^{\frac{4}{3}}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemerédi-Trotter point-line incidence bound.

Notes from several years of effort that did not yield a theorem, but might be of independent interest. 6 figures