Arithmetic Progressions, Nonunique Factorization, and Additive Combinatorics in the Group of Units Mod $n$

Abstract: For integers $0\lt a\leq b$, the arithmetic progression $M_{a,b}=a+b\mathbb{N}$ is closed under multiplication if and only if $a^2\equiv a \mod b$. Any such multiplicatively closed arithmetic progression is called an arithmetic congruence monoid (ACM). Though these $M_{a,b}$ are multiplicative submonoids of $\mathbb{N}$, their factorization properties differ greatly from the unique factorization one enjoys in $\mathbb{N}$.

In this talk we will explore the known factorization properties of these monoids. When $a=1$, these monoids are Krull and behave similarly to algebraic number rings, in that they have a class group which controls all the factorization. Combinatorially, factorization properties correspond to zero-sum sequences in the group. However, when $a\gt 1$, these monoids are not Krull and thus do not have a class group which fully captures the factorization behavior. Nonetheless, an ACM can be associated to a finite abelian group, whose additive combinatorics relate to the factorization properties of the ACM. We will pay particular attention to the factorization property of elasticity and its connection to sequences in the group which attain certain sums while avoiding others.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)