Point configurations determined by dot products

Abstract: Erdős' unit distance problem has perplexed mathematicians for decades. It asks for upper bounds on how often a fixed distance can occur in a large finite point set in the plane. We offer novel bounds on a family of variants of this problem involving multiple points, and relationships determined by dot products. Specifically, given a large finite set $E$ of points in the plane, and a $(m \times m)$ matrix $M$ of real numbers, we offer bounds on the number of $m$-tuples of points from $E$, $(x_1, x_2, \dots, x_m),$ satisfying $x_i \cdot x_j = m_{ij},$ the $(i,j)$th entry of $M$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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