Let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \log \log N$ prime factors. We may consider, say, $D = (\log \log N)^{1-\epsilon}$.
We do have rather precise approximations for the size of $Y$ (I am aware of Ch. II.6 in Tenenbaum's book, and references therein). I am wondering what work there is available on the Fourier transform $\widehat{1_Y}$ of the characteristic function of $Y$.
Comment: I would expect $\widehat{1_Y}$ to have peaks at the major arcs (i.e., arcs around rationals $a/q$ with small denominator). This is so because $Y$ is obviously "biased towards divisibility", and so should be somewhat overrepresented in the congruence class $0$ mod $d$ for any given $d$, relative to other congruence classes mod $d$. A back-of-the-envelope calculation suggests that the value at the peak around $a/q$ should be roughly proportional to $c^{\omega(q)}/q$. But what is known?
I am also interested in what happens when we let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \Delta$ prime factors in an interval $H<p\leq H^\Delta$, where $\Delta\to \infty$ as $N\to \infty$, $H\ggg 1$ and $H^\Delta\lll N$. This variant should be easier.
