SUM AND DIFFERENCE SETS IN GENERALIZED QUATERNION GROUPS
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Given a group πΊ, we say that a set π΄ β πΊ has more sums than differences (MSTD) if |π΄ + π΄| > |π΄ β π΄|, has more differences than sums (MDTS) if |π΄ + π΄| < |π΄ β π΄|, or is balanced if |π΄ + π΄| = |π΄ β π΄|. A problem of recent interest has been to understand the frequencies of these types of subsets. It is known that for arbitrary finite groups πΊ, almost all subsets π΄ β πΊ are balanced sets as |πΊ| β β. Recently for the generalized dihedral groups π· = β€β Γ πΊ, it is conjectured that there are more MSTD sets than MDTS sets. In this paper, we investigate the behavior of the sum and difference sets of π΄ β πββ, where πββ denotes generalized quaternion groups and show that the generalized quaternion group πββ has at least 2Β²βΏ subsets which are MSTD. We also analyze the expectation for |π΄ β π΄| where π΄ β πββ, proving an explicit formula for |π΄ β π΄| when π is prime.
Abstract
Given a group πΊ, we say that a set π΄ β πΊ has more sums than differences (MSTD) if |π΄ + π΄| > |π΄ β π΄|, has more differences than sums (MDTS) if |π΄ + π΄| < |π΄ β π΄|, or is balanced if |π΄ + π΄| = |π΄ β π΄|. A problem of recent interest has been to understand the frequencies of these types of subsets. It is known that for arbitrary finite groups πΊ, almost all subsets π΄ β πΊ are balanced sets as |πΊ| β β. Recently for the generalized dihedral groups π· = β€β Γ πΊ, it is conjectured that there are more MSTD sets than MDTS sets. In this paper, we investigate the behavior of the sum and difference sets of π΄ β πββ, where πββ denotes generalized quaternion groups and show that the generalized quaternion group πββ has at least 2Β²βΏ subsets which are MSTD. We also analyze the expectation for |π΄ β π΄| where π΄ β πββ, proving an explicit formula for |π΄ β π΄| when π is prime.
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