Most Subsets Are Balanced in Finite Groups

Abstract

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach’s conjecture and Fermat’s last theorem) can be formulated in terms of the sumset S + S = { x + y: x, y ∈ S} of a set of integers S. A finite set of integers A is sum-dominant if | A + A |  >  | A − A | . Though it was believed that the percentage of subsets of $\{0,\ldots,n\}$ that are sum-dominant tends to zero, in 2006 Martin and O’Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through the work of Zhao we know this percentage is approximately 4. 5 ⋅ 10⁻⁴). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that | A + A |  =  | A − A | tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of $\{0,\ldots,n\}$ have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.