Analytic Erdős-Turán conjectures and Erdős-Fuchs theorem (Q2368498)

Let \(A\) be a subset of the natural numbers with the property that every natural number can be represented as the sum of two elements of \(A\). It was conjectured by Erdős and Turán that the number of such representations of \(n\), denoted by \(r(A,n)\), is unbounded as \(n \to \infty\) and that \(| \sum_{k=0}^n (r(A,k)-c)| \) is unbounded as \(n \to \infty\) for every \(c>0\). In the latter direction, \textit{P. Erdős} and \textit{W. H. J. Fuchs} [J. Lond. Math. Soc. (2) 31, 67--73 (1956; Zbl 0070.04104)] showed that this sum cannot be \(o(n^{1/4}(\log n)^{-1/2})\). Call \(f=\sum_{n \geq 0} a_n X^n \in {\mathbb R}[[X]]\) a supported power series if there exists \(\gamma > 0\) such that for each \(n\) one has \(a_n=0\) or \(a_n > \gamma\). Further write \(r(f,n)=\sum_{i+j=n}a_ia_j\) for the coefficient of \(X^n\) in the series \(f^2\), so that \(r(f,n)\) coincides with \(r(A,n)\) when \(a_n\) is the indicator function of the set \(A\). The authors extend the Erdös-Fuchs Theorem to this more general situation, but with a slightly weaker conclusion: If \(f\) is a supported series and \(\sum_{k=0}^n(r(f,k)-c) = O(n^t)\) for some \(c>0\), then one has \(t \geq 1/4\). The proof is based on the method of \textit{D. J. Newman} [Proc. Am. Math. Soc. 75, No. 2, 209--210 (1979; Zbl 0418.10053)].