  We estimate Weyl sums over the integers with sum of binary digits either
fixed or restricted by some congruence condition. In our proofs we use ideas
that go back to a paper by Banks, Conflitti and the first author (2002).
Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem
which has been established by Bourgain, Demeter and Guth (2016) as well as by
Wooley (2016, 2019). We use our result to give an estimate of the discrepancy
of point sets that are defined by the values of polynomials at arguments having
the sum of binary digits restricted in different ways.
