On the image of convolutions along an arithmetic progression

Abstract: We consider the question of determining the structure of the set of all $d$-dimensional vectors of the form $N^{-1}(1_A*1_{-A}(x_1),..., 1_A*1_{-A}(x_d))$ for $A \subseteq {1,...,N}$, and also the set of all $(2N+1)^{-1}(1_B*1_B(x_1),..., 1_B*1_B(x_d))$, for $B \subseteq {-N, -N+1, ..., 0, 1, ..., N}$, where $x_1,...,x_d$ are fixed positive integers (we let $N \to \infty$). Using an elementary method related to the Birkhoff-von Neumann theorem on decompositions of doubly-stochastic matrices we show that both the above two sets of vectors roughly form polytopes; and of particular interest is the question of bounding the number of corner vertices, as well as understanding their structure.