On the image of convolutions along an arithmetic progression

Ernie Croot, Chi-Nuo Lee

We consider the question of determining the structure of the set of all $d$-dimensional vectors of the form $N^{-1}(1_A\ast 1_{-A}(x_1),...,1_A\ast 1_{-A}(x_d))$ for $A \subseteq \{1,...…