Convexity, superquadratic growth, and dot products

Abstract

Let (Formula presented.) be a point set with cardinality (Formula presented.). We give an improved bound for the number of dot products determined by (Formula presented.), proving that (Formula presented.) 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 (Formula presented.), there exist (Formula presented.) such that (Formula presented.) 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 Hanson, Roche-Newton, and Rudnev [Combinatorica, to appear], and uses predominantly elementary methods.

Department(s)

Mathematics

Document Type

Article

DOI

10.1112/jlms.12728

Publication Date

5-1-2023

Journal Title

Journal of the London Mathematical Society

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