Discrete Analysis Seminar

Let L_1 and L_2 be linear transformations from Z^d to Z^d satisfying certain mild conditions. Then, for any finite subset A of Z^d, we prove a lower bound for |L_1(A) + L_2(A)|, inspired from the Brunn-Minkowski inequality. This partially confirms a (corrected version of a) conjecture of Bukh and is best possible up to the lower-order term for many choices of L_1 and L_2. As an application, we prove a lower bound for |A + c.A| when A is a finite set of real numbers and c is an algebraic number.