Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3is an integer and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a power of 2. Furthermore, cannot be improved.

Given a commutative semigroup (S, +) with identity 0 and u × v matrices A and B with nonnegative integers as entries, we show that if C = A – B satisfies Rado's columns condition over ℤ, then any central set in S contains solutions to the system of equations . In particular, the system of equations is then partition regular. Restricting our attention to the multiplicative semigroup of positive integers (so that coefficients become exponents) we show that the columns condition over ℤ is also necessary for the existence of solutions in any central set (while the distinct notion of the columns condition over Q is necessary and sufficient for partition regularity over ℕ\{1}).