Sums and differences of correlated random sets

Text. Many questions in additive number theory (Goldbach's conjecture, Fermat's Last Theorem, the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair contributes one sum and two differences, we expect vertical bar A - A vertical bar > A + A vertical bar for finite sets A. However, Martin and O'Bryant showed a positive proportion of subsets of {0,, it) are sum-dominant. We generalize previous work and study sums and differences of pairs of correlated sets (A, B) (alpha is an element of {0,..., n} is in A with probability p, and a goes in B with probability pi if a E A and probability p(2) if alpha is not an element of A). If vertical bar A + B vertical bar > vertical bar(A - B) boolean OR (B - A)vertical bar, we call (A, B) a sum-dominant (1), P-2)-pair. We prove for any fixed (p) over bar = (p, p(1), p(2)) in (0,1)(3), (A, B) is a sum-dominant P2)-pair with positive probability, which approaches a limit p((p) over bar). We investigate p decaying with n, generalizing results of Hegarty-Miller on phase transitions, and find the smallest sizes of MSTD pairs. (C) 2014 Elsevier Inc. All rights reserved.