Treffer: Probabilistic study of sums of \(s\) \(s\)-th powers

By selecting the integers \(n\) independently, with probability \(1/ (sn^{1- 1/s})\), into a random set \(A\), one obtains a random analog of the \(s\)-th powers. Let \(R_n\) denote the number of representations of \(n\) in the form \(n= a_1+ \dots+ a_s\), with \(a_1< \dots< a_s\), \(a_i\in A\). The following properties of \(R_n\) are established: 1) \(R_n\) converges in distribution to a Poisson law with parameter \(\lambda= \Gamma (1/ s)^s s^{-s} s!^{-1}\). 2) With probability 1 the set of integers such that \(R_n =r\) has density \(e^{-\lambda} \lambda^r/ r!\) (in particular, the density of integers that have at least one representation is \(1- e^{-\lambda}\)). 3) The average number of representations tends to \(\lambda\). These results were announced without proof by \textit{P. Erdös} and \textit{A. Rényi} [Acta Arith. 6, 83-110 (1960; Zbl 0091.044)]. Several authors observed that an exact proof presents considerable difficulties due to the lack of independence. Here a detailed proof is given. The proof is based on a certain quasi-independence property. This is similar to, but in typical situations weaker than Janson's inequality [see \textit{S. Janson} et al., in: Random graphs '87, Proc. 3rd Int. Semin., Poznan/Poland 1987, 73-87 (1990; Zbl 0733.05073) and \textit{R. Boppona} and \textit{J. Spencer}, J. Comb. Theory, Ser. A 50, 305-307 (1989; Zbl 0663.60007)].