Abstract
For a nonempty set A of integers and an integer n, let rA(n) be the number of representations of n=a+a′ with a≤a′ and a,a′∈A, and let dA(n) be the number of representations of n=a−a′ with a,a′∈A. Erdős and Turán (1941) posed the profound conjecture: if A is a set of positive integers such that rA(n)≥1 for all sufficiently large n, then rA(n) 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 rA(n)=1 for all integers n. In this paper, we prove that, for any positive function l(x) with l(x)→0 as x→∞, there is a bounded set A of integers such that rA(n)=1 for all integers n and dA(n)=1 for all positive integers n, and A(−x,x)≥l(x)logx for all sufficiently large x, where A(−x,x) is the number of elements a∈A with −x≤a≤x.
Published Version
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