When Almost All Sets are Difference Dominated

Journal article, 2009

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,...,N}, chosen randomly according to a binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the random subset is almost surely difference dominated, as N --> oo, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums and differences in the random subset are almost surely distinct, and in particular the difference set is almost surely about twice as large as the sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_{f,g} N^{-1/2}, for some computable constant c_{f,g}, such that one form almost surely dominates below the threshold, and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p.