= Fourier separation of interval cutoffs
{c}
{title2=$1_{I_m}(n)=\int_0^1\widehat{1_{I_m}}(\theta)e(n\theta)\,d\theta$}
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 $H_N(\theta)\ll\min(N,\|\theta\|^{-1})$, 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 $mn\le N$.
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