Using the subspace theorem to bound unit distances

Abstract

We prove a special case of Erd\H{o}s' unit distance problem using a corollary of the subspace theorem bounding the number of solutions of linear equations from a multiplicative group. We restrict our attention to unit distances coming from a multiplicative group of rank $r$ not too large. Specifically, given $\varepsilon>0$ and $n$ points in the plane, we construct the unit distance graph from these points and distances and use the corollary above to bound certain paths of length $k$ in the graph giving at most $n^{1+\varepsilon}$ unit distances from the group above. We require that the rank $r\le c\log n$ for some $c>0$ depending on $\varepsilon$. This extends a result of J\'ozsef Solymosi, Frank de Zeeuw and the author where we only considered unit distances that are roots of unity. Lastly we show that the lower bound configuration for the unit distance problem of Erd\H{o}s consists of unit distances from a multiplicative subgroup of the form above.