The number of multiplicative Sidon sets of integers

Abstract

A set S of natural numbers is multiplicative Sidon if the products of all pairs in S are distinct. Erdős in 1938 studied the maximum size of a multiplicative Sidon subset of {1, . . . , n}, which was later determined up to the lower order term: π(n) + Θ( n 3/4 (log n) 3/2 ). We show that the number of multiplicative Sidon subsets of(log n) 3/2 ) for a certain function T (n) ≈ 2 1.815π(n) which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determined. It resolves the enumeration problem for multiplicative Sidon sets initiated by Cameron and Erdős in the 80s.We also investigate its extension for generalised multiplicative Sidon sets. Denote by S k , k ≥ 2, the number of multiplicative k-Sidon subsets of {1, . . . , n}. We show that S k (n) = (β k + o(1)) π(n) for some β k we define explicitly. Our proof is elementary.1

Sidon sets for linear forms

“…For surveys of classical Sidon sets, see Halberstam and Roth [2] and O'Bryant [7]. For recent work, see [1,3,4,5,6,8,9,11,13].…”

Integer Colorings with No Rainbow 3-Term Arithmetic Progression

“…In the setting of integers, this problem was introduced by Cameron and Erdős [11] who studied the number of subsets of positive integers satisfying some constraint. We refer to [5,18,28] for the number of sum-free subsets, to [4] for the number of subsets with no k-term arithmetic progression, and to [23] for results on multiplicative Sidon sets.…”

Integer colorings with no rainbow 3-term arithmetic progression

“…In the setting of integers, this problem was introduced by Cameron and Erdős [11] who studied the number of subsets of positive integers satisfying some constraint. We refer to [5] for the number of sum-free subsets, to [4] for the number of subsets with no k-term arithmetic progression, and to [22] for results on multiplicative Sidon sets.…”