Diophantic Inequality. davenport Theorem

Question

I'm studying the topic sums over primes, but I had a problem when studying the outcome of Davenport, $$\sum_{n\leqslant x} \mu(n) e(\alpha n) = O(x(\log x)^{-A})$$ more exactly, in a diophantic inequality of Baker $$\sum_{|m|\leqslant M} \min \{N, \frac{1}{2||\alpha m||} \} \leqslant (1 +4M q^{-1})\big( N + \sum_{1\leqslant l \leqslant q} ql^{-1} \big)$$ where the point $||\alpha m||$ are all distinct and spaced by $1/2q$ at least. I prove a inequality but I did not get the same expression, by using a inequality $$ \sum_{h=1}^H \min \{N,\frac{1}{||\alpha h||} \} \leqslant \sum_{k=0}^{[2H/q]} \sum_{h=k[q/2]+1}^{(k+1)[q/2]} \min \{N,\frac{1}{||\alpha h||} \} $$ Any suggestions?