A Structural Condition on Point Sets with Few Distinct Dot Products

The distinct dot products problem, a variant of the Erdős distinct distances problem, asks “Given a set $P_n$ of $n$ points in ${\mathbb{R}}^2$, what is the minimum number $\mathrm{\mid }D(P_n)\mathrm{\mid }$ of distinct dot products they determine?” The best proven lower bound is $\mathrm{\mid }D(P_n)\mathrm{\mid }=\mathrm{\Omega }(n^{2\mathrm{/}3+7\mathrm{/}1425})$, due to work by Hanson$\text{unicode}x2013$Roche-Newton$\text{unicode}x2013$Senger, and a recent improvement by Kokkinos. However, the best known construction determines $\mathrm{\Theta }(n)$ dot products. We provide a structural condition that a point configuration $P_n$ would have to satisfy in order to have ‘few’ dot products, by which we mean that $\mathrm{\mid }D(P_n)\mathrm{\mid }<n^{\frac{3}{4}(1-\epsilon )}$ for some $\epsilon >0$.