Equidistribution conditions for gaps of geometric numerical semigroups

Authors

Caleb Shor, Western New England UniversityFollow
Jae Hyung Sim

College

College of Arts and Sciences

Department

Mathematics

Publication Date

11-7-2025

Abstract

In 2008, Wang & Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers a and b is equidistributed modulo 2 precisely when a and b are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo m when a and b are coprime to m and at least one of them is 1 modulo m. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form a^k, a^k-1b,...,b^k, aiming to determine when the corresponding set of gaps is equidistributed modulo m. With elementary methods, we are able to obtain a result for k = 2 and all m. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all k and infinitely many m. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all k and all m.

Recommended Citation

Shor, Caleb and Sim, Jae Hyung, "Equidistribution conditions for gaps of geometric numerical semigroups" (2025). Faculty Publications - College of Arts and Sciences. 11.
https://digitalcommons.law.wne.edu/casfacpubs/11