An improved bound on the number of dot products determined by a finite point set in the plane

TL;DR

The paper addresses the problem of lower-bounding the number of distinct dot products determined by a finite planar point set $P$, proving $|\Lambda(P)| \gtrsim |P|^{\frac{2}{3}+\frac{7}{1425}}$. It combines additive-combinatorics tools (Plünnecke–Ruzsa inequalities, multiplicative energy, and energy–intersection bounds) with geometric incidence bounds (Elekes–Szabó framework and Szemerédi–Trotter) to derive a superquadratic expander effect and a dyadic structural decomposition of $P$. The core argument uses lines through the origin and pins a pair of slopes to produce large intersection sets $Z$, relating them to dot-product counts via two auxiliary sets $\Lambda_1,\Lambda_2$, and then applies a Plünnecke-type inequality to bound $|\Lambda(P)|$ in terms of a large expander quantity. The optimal choice of parameters yields the exponent $\frac{2}{3}+\frac{7}{1425}$, improving the constant beyond $\tfrac{2}{3}$ and advancing parallels with the distinct distances problem in the plane.

Abstract

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the improved lower bound \[|\{p\cdot q : p,q\in P\}|\gtrsim |P|^{\frac2 3 + \frac{7}{1425}}.\]

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5