We investigate subsets A of the natural numbers having the property that, for some positive number p < 2, one has
int_0^1 | sum_{n in A\cap [1,N]} e(n alpha) |^p d alpha
<< | A\cap [1,N] |^p N^{eps-1}.
Examples of such sets include (but are not restricted to) the square-free, or more generally, the r-free numbers. We show that there are many other examples of such sets. For polynomials
psi(x; a) = a _kx^k + … + a_1x,
having coefficients a_i satisfying suitable irrationality conditions, we obtain Weyl-type estimates for associated exponential sums restricted to subconvex L^p sets, and we show that the sequence (psi(n; a))_{n in A} is equidistributed modulo 1. We also discuss applications to averages of arithmetic functions.