Multiplicative Properties of Hilbert Cubes

TL;DR

This paper establishes upper bounds on the size of Hilbert cubes in finite fields that avoid large product sets and reciprocals of sum sets, using character sum bounds, advancing understanding of their multiplicative structure.

Contribution

It provides new upper bounds on Hilbert cubes in finite fields avoiding certain multiplicative configurations, using a different approach from prior estimates.

Findings

Upper bounds on Hilbert cubes avoiding large product sets

Bounds on Hilbert cubes avoiding reciprocals of sum sets

Application of character sum bounds to combinatorial structures

Abstract

We obtain upper bounds on the cardinality of Hilbert cubes in finite fields, which avoid large product sets and reciprocals of sum sets. In particular, our results replace recent estimates of N. Hegyv\'ari and P. P. Pach (2020), which appear to be void for all admissible parameters. Our approach is different from that of N. Hegyv\'ari and P. P. Pach and is based on some well-known bounds of double character and exponential sums over arbitrary sets, due to A. A. Karatsuba (1991) and N. G. Moshchevitin (2007), respectively.