Linear phase

A linear phase on an abelian group is an additive character, such as $x\mapsto e(\theta x)$ on the integers or $x\mapsto {\omega }^{rx}$ on ${\mathbb{Z}}_N$.

For a linear phase $e^{2\pi i\theta x}$ on $[n]$, set $Q=\lfloor \sqrt{n}\rfloor$. The Dirichlet approximation theorem gives $1\le q\le Q$ with $\Vert q\theta {\Vert }_{\mathbb{R}/\mathbb{Z}}\le Q^{-1}$. If $1\le L$ and $2L\le Q$, split each residue chain modulo $q$ into arithmetic progressions of lengths between $L$ and $2L-1$, merging the final remainder into the preceding block. Each chain has at least $Q$ points, so this is possible. On each resulting arithmetic progression, the linear phase varies from its first value by at most $4\pi L/Q$. This converts a large Fourier coefficient of a zero-sum real function into positive mean on one block: freezing the linear phase controls the weighted block sums, and their positive and negative total masses agree.