A Structural Condition on Point Sets with Few Distinct Dot Products

The distinct dot products problem, a variant of the Erd\H{o}s distinct distances problem, asks "Given a set $P_n$ of $n$ points in $\mathbb{R}^2$, what is the minimum number $|D(P_n)|$ of distinct dot products they determine?" The best proven lower bound is $|D(P_n)| = \Omega(n^{2/3+7/1425})$, due to work by Hanson$\unicode{x2013}$Roche-Newton$\unicode{x2013}$Senger, and a recent improvement by Kokkinos. However, the best known construction determines $\Theta(n)$ 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 $|D(P_n)| < n^{\frac{3}{4}(1-\epsilon)}$ for some $\epsilon > 0$.