Equidistribution Conditions for Gaps of Geometric Numerical Semigroups

2025/03/13 by Shor, Caleb M., Sim, Jae Hyung
#11D04 #11F66 #11F70 #11J82 #11R18 #20M14 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.10826

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.

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