Bounded unique representation bases for the integers

Abstract

For a nonempty set $A$ of integers and an integer $n$, let $r_A\left(n\right)$ be the number of representations of $n=a+a^{\prime }$ with $a\le a^{\prime }$ and $a,a^{\prime }\in A$, and let $d_A\left(n\right)$ be the number of representations of $n=a-a^{\prime }$ with $a,a^{\prime }\in A$. Erdős and Turán (1941) posed the profound conjecture: if $A$ is a set of positive integers such that $r_A\left(n\right)\ge 1$ for all sufficiently large $n$, then $r_A\left(n\right)$ is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set $A$ with logarithmic growth such that $r_A\left(n\right)=1$ for all integers $n$. In this paper, we prove that, for any positive function $l\left(x\right)$ with $l\left(x\right)\to 0$ as $x\to \infty$, there is a bounded set $A$ of integers such that $r_A\left(n\right)=1$ for all integers $n$ and $d_A\left(n\right)=1$ for all positive integers $n$, and $A(-x,x)\ge l\left(x\right)logx$ for all sufficiently large $x$, where $A(-x,x)$ is the number of elements $a\in A$ with $-x\le a\le x$.