Weyl sums over integers with affine digit restrictions

For any given integer q⩾2, we consider sets N of non-negative integers that are defined by affine relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1's equals twice the number of 0's in the binary representation). The main goal is to prove that the sequence (αn)n∈N is uniformly distributed modulo 1 for all irrational numbers α. The proof is based on a saddle point analysis of certain generating functions that allows us to bound the corresponding Weyl sums.