Multiplicative Properties of Hilbert Cubes
Number Theory
2022-03-15 v2 Combinatorics
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.
Keywords
Cite
@article{arxiv.2201.00585,
title = {Multiplicative Properties of Hilbert Cubes},
author = {Igor E. Shparlinski},
journal= {arXiv preprint arXiv:2201.00585},
year = {2022}
}