Convexity, Superquadratic Growth, and Dot Products

TL;DR

This paper improves bounds on the number of distinct dot products in a point set and introduces new superquadratic expanders involving products and shifts, utilizing elementary methods and growth set analysis.

Contribution

It provides an improved lower bound for dot product counts and introduces a novel superquadratic expander involving products and shifts, advancing additive combinatorics techniques.

Findings

Bound for dot products:  | p  q : p,q  P    N^{2/3+c}

Existence of z,z' in X with superquadratic growth in set expansions

Development of elementary methods for growth set analysis

Abstract

Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set $X \subset \mathbb R$, there exist $z,z' \in X$ such that \[ \left|\frac{(zX+1)^{(2)}(z'X+1)^{(2)}}{(zX+1)^{(2)}(z'X+1)}\right| \gtrsim |X|^{5/2}. \] This is derived from a more general result concerning growth of sets defined via convexity and sum sets, and which can be used to prove several other expanders with better than quadratic growth. The proof develops arguments from recent work by the first two listed authors and Misha Rudnev, and uses predominantly elementary methods.