Equidistribution Conditions for Gaps of Geometric Numerical Semigroups

TL;DR

This work provides a complete framework for understanding when the gaps of geometric numerical semigroups $S=\langle \mathrm{Geom}(a,b;k)\rangle$ are equidistributed modulo an integer $m$. Building on polynomial-congruence methods that reduce equidistribution to identities modulo $C_m(x)$, the authors first settle the $k=2$ case with explicit modular conditions, then harness cyclotomic-unit theory and Dirichlet $L$-functions to treat arbitrary $k$ and all $m$. The key innovations include (i) translating gap-structure into exponent-congruence problems, (ii) deploying cyclotomic-unit dependence and Stark-type relations to bound possibilities, and (iii) deriving a seven-condition main theorem that precisely governs equidistribution in terms of $a,b,k,m$ and their congruence classes. The results illuminate the deep arithmetic connections between numerical semigroups, cyclotomic fields, and special values of $L$-functions, with implications for understanding equidistribution phenomena in broader families of semigroups.

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

Figures (1)

• Figure 1: Reductions of $L(x)$ and $R(x)$ modulo $(x^m-1)$, from Example \ref{['ex:checking-equidist-k=2']}

Theorems & Definitions (59)

• Proposition 1.1: WangWang2008
• Proposition 1.2: Shor22
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5: Shor22
• Remark 2.6
• Proposition 2.7
• Proposition 2.8