= One-frequency density increment on an integer interval
{title2=$|P|\ge c\alpha^2\sqrt N,\quad |A\cap P|/|P|\ge\alpha+\alpha^2/144$}
For an <indicator function> of a subset of $[N]$ with <subset density> $\alpha$, a large <Fourier coefficient> of $1_A-\alpha1_{[N]}$ gives a <density increment> on an <arithmetic progression>. The <Dirichlet approximation theorem> supplies a step $d\le\sqrt N$ on which the detected <linear phase> changes slowly. Split its residue classes into progressions of lengths comparable to $\alpha^2\sqrt N$. Replacing the phase by one constant on each cell preserves a fixed fraction of the Fourier correlation. Since the balanced function has total sum zero, positive discrepancies sum to half the total absolute discrepancy; one cell has the displayed increased density. One may take $c=1/(2304\pi)$ when $N$ is sufficiently large in terms of $\alpha$.
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