 Fourier separation of interval cutoffs

If $I_m$ is an integer interval inside $[1,N]$, discrete Fourier inversion gives the displayed identity. The exponential geometric sum bound gives a common envelope $H_N(\theta )\ll \min (N,\Vert \theta {\Vert }^{-1})$, whose integral is $O(\log (2N))$. At each frequency, $\widehat{1_{I_m}}(\theta )/H_N(\theta )$ is a bounded coefficient depending only on $m$, while $e(n\theta )$ depends only on $n$. A uniform bounded-coefficient Type II sum estimate therefore survives an $m$-dependent interval cutoff at a cost of one logarithm. This is particularly useful for the condition $mn\le N$.