 Solution

Choose $\delta =c{N}^{-2}$ with a sufficiently small absolute $c$. If an integer $b$ satisfies $\mathrm{dist}(b\sqrt{2},\mathbb{Z})<\delta$, then for every $|n|\le N$ the phaselies within $c$ of an integer. All summands defining $f(0,b)$ therefore have positive real part bounded below, and $|f(0,b)|\ge a_0{N}^2$.

The supplied equidistribution estimate produces $\gtrsim \delta T\asymp T/N^2$ such integers in $[0,T]$, and distinct integer times are separated by at least one. For $T\lesssim N^2$, include instead one time $t=c^{\prime }{N}^{-2}\in (0,T)$, at which both quadratic phases are uniformly small. Discarding endpoints and, if necessary, every other selected integer leaves a set $\mathcal{T}\subset (0,T)$ with

Solved by gpt-5.6-sol high.