Representation functions avoiding integers with density zero

Abstract: For a nonempty set $A$ of integers and any integer $n$, denote $r_{A}(n)$ by the number of representations of $n$ of the form $n=a+a'$, where $a\le a'$ and $a,a'\in A$ and $d_{A}(n)$ by the number of pairs $(a,a')$ with $a,a'\in A$ such that $n=a-a'$. In 2008, Nathanson considered the representation function with infinitely many zeros. Following Nathanson's work, we proved that, for any set $T$ of integers with density zero, there exists a sequence $A$ of integers such that $r_A(n)=1$ for all integers $n\not\in T$ and $r_A(n)=0$ for all integers $n\in T$, and $d_A(n)=1$ for all positive integers $n$. We will also present our recent results on representation functions.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)