A note on product sets of random sets
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A NOTE ON PRODUCT SETS OF RANDOM SETS C. SANNA Department of Mathematics, Universit` a di Genova, Genova, Italy e-mail: [email protected] (Received August 21, 2019; revised October 28, 2019; accepted October 29, 2019)
Abstract. Given two sets of positive integers A and B, let AB := {ab : a ∈ A, b ∈ B} be their product set and put Ak := A · · · A (k times A) for any positive integer k. Moreover, for every positive integer n and every α = α(n) ∈ [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 if A1 , . . . , As are random sets in B(n1 , α1 ), . . . , B(ns , αs ), respectively, k1 , . . . , ks are fixed positive integers, αi ni → +∞, and 1/αi does not k1
ks
grow too fast in terms of a product of log nj ; then |Ak1 1 · · · Aks s | ∼ |Ak11|! · · · |Akss|! with probability 1 − o(1). This is a generalization of a result of Cilleruelo, Ramana, and Ramar´e [3], who considered the case s = 1 and k1 = 2.
1. Introduction Given two sets of positive integers A and B, let AB := {ab : a ∈ A, b ∈ B} be their product set and put Ak := A · · · A (k times A) for any positive integer k. Problems involving the cardinalities of product sets have been considered by many researchers. For example, the study of Mn := |{1, . . . , n}2 | as n → +∞ is known as the “multiplicative table problem” and was started by Erd˝os [5,6]. The exact order of magnitude of Mn was determined by Ford [7] following an earlier work of Tenenbaum [12]. Furthermore, Koukoulopoulos [10] provided uniform bounds for |{1, . . . , n1 } · · · {1, . . . , ns }| holding for a wide range of n1 , . . . , ns . Cilleruelo, Ramana, and Ramar´e [3] proved asymptotics or bounds for |(A ∩ {1, . . . , n})2 | when A is the set of shifted prime numbers, the set of sums of two squares, or the set of shifted sums of two squares. C. Sanna is supported by a postdoctoral fellowship of INdAM and is a member of the INdAM group GNSAGA. Key words and phrases: asymptotics, product set, random set. Mathematics Subject Classification: primary 11N37, secondary 11N99. c 2019 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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C. C. SANNA SANNA
For every positive integer n and every α = α(n) ∈ [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. Number-theoretic problems involving this probabilistic model have been considered by several authors [1–4,11]. In particular, Cilleruelo, Ramana, and Ramar´e [3] proved the following: Theorem 1.1. Let A be a random set in B(n, α). If αn → +∞ and 2 α = o((log n)−1/2 ), then |A2 | ∼ |A| 2 with probability 1 − o(1). The contribution of this paper is the following generalization of Theorem 1.1. Theorem 1.2. Let A1 , . . . , As be random sets in B(n1, α1 ), . . . , B(ns, αs ), respectively; and let k1
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