Abstract
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 \(A_1\), \(\ldots ,\) \( A_s\) 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, \(\alpha _i n_i \rightarrow +\infty \), and \(1/\alpha _i\) does not grow too fast in terms of a product of \(\log n_j\); then \(|A_1^{k_1} \cdots A_s^{k_s}| \sim \frac{|A_1|^{k_1}}{k_1!}\cdots \frac{|A_s|^{k_s}}{k_s!}\) with probability \(1 - o(1)\). This is a generalization of a result of Cilleruelo, Ramana, and Ramaré [3], who considered the case \(s = 1\) and \(k_1 = 2\).
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C. Sanna is supported by a postdoctoral fellowship of INdAM and is a member of the INdAM group GNSAGA.
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Sanna, C. A note on product sets of random sets. Acta Math. Hungar. 162, 76–83 (2020). https://doi.org/10.1007/s10474-019-01014-4
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DOI: https://doi.org/10.1007/s10474-019-01014-4