On the distribution of sums of residues

We generalize and solve the ${\bmod \;q}$ analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the ${2^n}$ sums of the form ${\sum _{i = 1}^n{\varepsilon _i}{a_i}}$, where each ${\varepsilon _i}$ is 0 or 1. For all q, n, k we determine the maximum, over all reduced residues ${a_i}$ and all sets P consisting of k arbitrary residues, of the number of these sums that belong to P.