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A note on product sets of random sets. (English) Zbl 1474.11164

Summary: Given two sets of positive integers \(A\) and \(B\), let \(AB := \{ab : a \in A, b\in B\}\) be their product set and put \(A^k := A \cdots A\) (\(k\) times \(A\)) for any positive integer \(k\). Moreover, for every positive integer \(n\) and every \(\alpha = \alpha(n)\in [0, 1]\), let \(\mathcal{B}(n, \alpha)\) denote the probabilistic model in which a random set \(A \subseteq \{1, \ldots,n\}\) is constructed by choosing independently every element of \(\{1, \ldots,n\}\) with probability \(\alpha\). We prove that if \(A_1,\ldots,A_n\) are random sets in \(\mathcal{B}(n_1, \alpha_1),\ldots,\mathcal{B}(n_s, \alpha_s)\), respectively, \(k_1,\ldots,k_s\) are fixed positive integers, \(\alpha_in_i\to +\infty\) and \(1/\alpha_i\) does not grow too fast in terms of a product of \(\log n_j\); then \(\vert A_1^{k_1}\cdots A_s^{k_s}\vert \sim \frac{\vert A_1\vert ^{k_1}}{k_1!}\cdots \frac{\vert A_s\vert ^{k_1}}{k_s!}\) with probability \(1-o(1)\). This is a generalization of a result of J. Cilleruelo et al. [J. Number Theory 144, 92–104 (2014; Zbl 1296.11007)], who considered the case \(s = 1\) and \(k_1 = 2\).

MSC:

11N37 Asymptotic results on arithmetic functions
11N99 Multiplicative number theory

Citations:

Zbl 1296.11007
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References:

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