Equidistribution Conditions for Gaps of Geometric Numerical Semigroups

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 $a^k, a^{k-1}b, \dots, b^k$, 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 $m$.