On the general Erdős-Turán conjecture (Q2019275)

Summary: The general Erdős-Turán conjecture states that if \(A\) is an infinite, strictly increasing sequence of natural numbers whose general term satisfies \(a_n\leq cn^2\), for some constant \(c\) and for all \(n\), then the number of representations functions of \(A\) is unbounded. Here, we introduce the function \(\psi(n)\), giving the minimum of the maximal number of representations of a finite sequence \(A=\{a_k\colon 1\leq k\leq n\}\) of \(n\) natural numbers satisfying \(a_k\leq k^2\) for all \(k\). We show that \(\psi(n)\) is an increasing function of \(n\) and that the general Erdős-Turán conjecture is equivalent to \(\lim_{n\to\infty}\psi(n)=\infty\). We also compute some values of \(\psi(n)\). We further introduce and study the notion of capacity, which is related to the \(\psi\)-function by the fact that \(\lim_{n\to\infty}\psi(n)\) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest. The present paper is a sequel to the authors' paper [Acta Arith. 117, No. 1, 81--105 (2005; Zbl 1142.11005)], to which we refer for broader background and further motivation.