One-frequency density increment on an integer interval ($|P|\ge c{\alpha }^2\sqrt{N},|A\cap P|/|P|\ge \alpha +{\alpha }^2/144$)

For an indicator function of a subset of $[N]$ with subset density $\alpha$, a large Fourier coefficient of $1_A-\alpha 1_{[N]}$ 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 $\alpha$.