Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Question

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle method (at least for the 'classical' proof I am aware of).

We know by Vaughn's identity that if $|\alpha - a/q| < 1/q^2$, $(a,q)=1$, $q \leq n$ then we have an estimate of the form $$ \sum_{p \leq n} (\log p )e(p \alpha) \ll (n q^{-1/2} + n^{4/5} + n^{1/2}q^{1/2}) (\log n )^4, $$ and in case when $(\log n)^{c_1} \leq q \leq n/(\log n)^{c_2}$ we obtain a non-trivial estimate for the above sum.

My question is: are there other examples of sequence say $M = \{m_1, m_2, .. \} \subseteq \mathbb{N}$ such that we have an estimate of the above type if we replace the primes with $M$. In other words, are there examples of sequences $M$ (and possibly with a suitable weight function $f(m)$ ) such that for $$ \sum_{m \in M, m \leq n} f(m) e(m \alpha) $$ we have a similar estimate as above where we can obtain a non-trivial estimate when $q$ is in the appropriate range?

Thank you very much!

Answer 1

Bruedern, Granville, Perelli, Vaughan and Wooley, (Philos. Trans. Roy. Soc. London Ser. A, 356 (1998) 739 - 761) dealt with the sequence of $k$-free integers. Bruedern (in: Analytic Number Theory, Cambridge University Press 2009, 91-132) generalized this approach to almost periodic sequences, as obtained by converging sieves. In both cases the nice thing is that, differently from the case of primes, the $L^2$-norm of the exponential sum is concentrated in the major arcs.