A generalization of a theorem of Erdős-Rényi to \(m\)-fold sums and differences (Q2925469)

A set \(S\subseteq\mathbb{N}\) of positive integers is called of type \(B_{2}(g)\) if for all \(n\in\mathbb{N}\) the number of representations of the form \(n=s_{1}+s_{2}\) (with \(s_{1}\leq s_{2}\)) is bounded from above by \(g\). \textit{P. Erdős} and \textit{A. Rényi} [Acta Arith. 6, 83--110 (1960; Zbl 0091.04401)] have shown that for any \(\varepsilon > 0\) there exists \(g=g(\varepsilon)\) and a set \(S\subseteq\mathbb{N}\) of type \(B_{2}(g)\) such that for sufficiently large \(n\) \(S(n)> n^{1/2-\varepsilon}\), where \(S(n)\) is the number of elements of \(S\) not exceeding \(n\). The result is optimal up to the \(\varepsilon\)-term in the exponent and the proof is of probabilistic nature. A detailed discussion is given in [\textit{H. Halberstam} and \textit{K. F. Roth}, Sequences. (Reprint). York-Heidelberg-Berlin: Springer-Verlag (1983; Zbl 0498.10001)]. \textit{V. H. Vu} [Duke Math. J. 105, No. 1, 107--134 (2000; Zbl 1013.11063)] extended this result to representations as to \(m-\)fold sums \(s_{1}+\ldots + s_{m}\) and the present paper gives a further generalization in this direction. The lower bound is of the form \(n^{1/m-\varepsilon}\).