  Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are
distinct integers. We call f an idempotent trigonometric polynomial (because
the convolution of f with itself is f) or, simply, an idempotent. We show that
for every p > 1, and every set E of the torus T = R/Z with |E| > 0, there are
idempotents concentrated on E in the Lp sense. More precisely, for each p > 1,
there is an explicitly calculated constant Cp > 0 so that for each E with |E| >
0 and epsilon > 0 one can find an idempotent f such that the pth root of the
ratio of the integral over E of the pth power of |f| to the integral over T of
the pth power of |f| is greater than Cp - epsilon. This is in fact a lower
bound result and, though not optimal, it is close to the best that our method
gives. We also give both heuristic and computational evidence for the still
open problem of whether the Lp concentration phenomenon fails to occur when p =
1.
