On bounded unique representation bases

Abstract: For a nonempty set $A$ of integers and an integer $n$,  let $r_{A}(n)$ be the number of representations of $n=a+a'$ with $a\le a'$ and $a, a'\in A$, and let $d_{A}(n)$ be the number of representations of $n=a-a'$ with $a, a'\in A$. In 1941, Erd\H{o}s and Tur\'{a}n posed the profound conjecture: If $A$ is a set of positive integers such that $r_A(n)\ge 1$ for all sufficiently large $n$, then $r_A(n)$ is unbounded. In 2004, Ne\v{s}et\v{r}il and Serra introduced the notion of bounded sets and confirmed the Erd\H{o}s-Tur\'{a}n conjecture for all bounded bases. In 2003, Nathanson considered the existence of the set $A$ with logarithmic growth such that $r_A(n)=1$ for all integers $n$. Recently, we prove that, for any positive function $l(x)$ with $l(x)\rightarrow 0$ as $x\rightarrow \infty$,  there is a bounded set $A$ of integers such that $r_A(n)=1$ for all integers $n$ and $d_A(n)=1$ for all positive integers $n$, and $A(-x,x)\ge l(x)\log x$ for all sufficiently large $x$, where $A(-x,x)$ is the number of elements $a\in A$ with $-x\le a\le x$. This is joint work with Prof. Yong-Gao Chen.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)