On Multiplicative Sidon Sets

Fix integers b > a ≥ 1 with g := gcd(a, b). A set S ⊆ N is {a, b}-multiplicative if ax 6= by for all x, y ∈ S. For all n, we determine an {a, b}-multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}-multiplicative set is b b+g . Erdős [2, 3, 4] defined a set S ⊆ N to be multiplicative Sidon1 if ab = cd implies {a, b} = {c, d} for all a, b, c, d ∈ S; see [8–10]. In a similar direction, Wang [13] defined a set S ⊆ N to be double-free if x 6= 2y for all x, y ∈ S, and proved that the maximum density of a double-free set is 23 ; see [1] for related results. Here the density of S ⊆ N is lim n→∞ |S ∩ [n]| n . Pór and Wood [7] generalised the notion of double-free sets as follows. For k ∈ N, a set S ⊆ N is k-multiplicative (Sidon) if ax = by implies a = b and x = y for all a, b ∈ [k] and x, y ∈ S. Pór and Wood [7] proved that the maximum density of a k-multiplicative set is Θ( 1 log k ). Here we study the following alternative generalisation of double-free sets. For distinct a, b ∈ N, a set S ⊆ N is {a, b}-multiplicative if ax 6= by for all x, y ∈ S. Our main result is to determine the maximum density of an {a, b}-multiplicative set. Assume that a < b throughout. Say x ∈ N is an i-th subpower of b if x = biy for some y 6≡ 0 (mod b). If x is an i-th subpower of b for some even/odd i then x is an even/odd subpower of b. First suppose that gcd(a, b) = 1. Let T be the set of even subpowers of b. We now prove that T is an {a, b}-multiplicative set with maximum density. In fact, for all [n], ∗Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia ([email protected]). Supported by a QEII Research Fellowship from the Australian Research Council. Additive Sidon sets have been more widely studied; see the classical papers [5, 11, 12] and the recent survey by O’Bryant [6]. Let N := {1, 2, . . . } and [n] := {1, 2, . . . , n}.