Monotone increasing representation functions

Abstract: Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. A few years ago, Bell and Shallit constructed a set $A$ of natural numbers such that $\mathbb{N}\setminus A$ is infinite, but the corresponding representation function is strictly increasing. Later, together with Csaba Sándor and Yang Quan-Hui, we improved their result. Furthermore, we constructed a dense set such that the corresponding representation function is not strictly increasing. In my talk I will also give an overview of the recent progress on this topic.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)