On maximal product sets of random sets

2020/05/10 by Mastrostefano, Daniele
#11B99 (Primary) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2005.04663

Abstract

For every positive integer N and every α∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A⊂ \1,…,N\ is constructed by choosing independently every element of \1,…,N\ with probability α. We prove that, as N\longrightarrow +∞, for every A in B(N, α) we have |AA| ∼ |A|²/2 with probability 1-o(1), if and only if \fraclog(α²(log N)^{log 4-1})√(loglog N)\longrightarrow-∞. This improves a theorem of Cilleruelo, Ramana and Ramaré, who proved the above asymptotic between |AA| and |A|²/2 when α=o(1/√(log N)), and supplies a complete characterization of maximal product sets of random sets.

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