On multiplicative sidon sets

David Wakeham, David R. Wood

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Abstract

Fix integers b > a ≥ 1 with g:= gcd(a, b). A set S ⊆ N is {a, b}-multiplicative if ax ≠ by for all x, y ϵ 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. For A, B ⊆ N, a set S ⊆ N is {A, B}-multiplicative if for all a ϵ A and b ϵ B and x, y ϵ S, the only solutions to ax = by have a = b and x = y. For 1 < a < b < c and a, b, c coprime, we give a O(1) time algorithm to approximate the maximum density of an {{a}, {b, c}}-multiplicative set to arbitrary given precision.

Original languageEnglish
Title of host publicationIntegers: Annual Volume 2013
PublisherWalter de Gruyter
Pages392-401
Number of pages10
ISBN (Electronic)9783110298161
ISBN (Print)9783110298116
DOIs
Publication statusPublished - 1 Jan 2014

Cite this

Wakeham, D., & Wood, D. R. (2014). On multiplicative sidon sets. In Integers: Annual Volume 2013 (pp. 392-401). Walter de Gruyter. https://doi.org/10.1515/9783110298161.392