ALMOST NO FINITE SUBSET OF INTEGERS CONTAINS A qTH POWER MODULO ALMOST EVERY PRIME

Let q be a prime. We give an elementary proof of the fact that for any k is an element of N, the proportion of k -element subsets of Z that contain a qth power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of k -element subsets of [-N, N]boolean AND Z that contain a qth power modulo almost every prime is no larger than aq,kNk-(1-1q), for some positive constant aq,k.Furthermore, the number of k -element subsets of {+/- pe1 1 pe2 2 center dot center dot center dot peNN : 0 5 e1, e2, . . . , eN 5 N} that contain a qth power modulo almost every prime is no larger than mq,k NNk qN for some positive constant mq,k.