Weyl sums in the arithmetic progressions

Question

For any $\alpha \in \mathbb{R}$ which has the Diophantine Approximation that $$\alpha=\frac{l}{q}+\frac{\theta}{q^2},\quad (l,q)=1, \quad|\theta|\le 1.$$ It is known that $$\sum_{m\le M} \min \left(N,\frac{1}{\|m\alpha\|} \right)\ll \left(1+\frac{M}{q} \right) \left(N+q\log N \right)$$ for any $M,N\ge 2$. I have a question which may be naive for the expects here; the puzzle is, for any $P\in \mathbb{N}^+$ such that $P\le M$, does one have the analogous estimates for the sum $$\sum_{\substack{m\le M\\m \equiv 0 \bmod P}} \min \left(N,\frac{1}{\|m\alpha\|} \right) ，\tag{1}$$ and, more generally, the sum $$\sum_{\substack{m\le M\\m \equiv a \bmod P}} \min \left(N,\frac{1}{\|m\alpha\|} \right)\tag{2}$$ for any fixed $a\in \mathbb{N}$? I searched certain papers, but cannot find any references exactly discussing about this. So any ideas or references are welcome.

Thanks in advance.

Answer 1

Suppse $\alpha$ has rational approximation $b/p$. The standard technique reduces to counting the cardinality of sets of the form

$$\{ m\le M, \ m\equiv a\mod{q} \ : \ |mb \mod{p}|\le \varepsilon p\}.$$

or

$$\{ m\le M/q \ : \ |mab+c \mod{p}|\le \varepsilon p\}.$$

for a certain integer $c$ for which there are $$\ll \left(1+\frac{M}{pq}\right)\varepsilon p$$ solutions.