Circle cap Fourier lower bound ($|\widehat{\psi d\sigma }|\gtrsim \delta \text{ on a }{\delta }^{-2}\times {\delta }^{-1}\text{ rectangle}$)

For normalized circle measure and a nonnegative cutoff near $(1,0)$, equal to one on $|\theta |\le \delta /C$ and supported on $|\theta |\le 2\delta /C$, remove the constant phase $e^{-i{\xi }_1}$. On $|{\xi }_1|\le {\delta }^{-2}$, $|{\xi }_2|\le {\delta }^{-1}$, the remaining phase has magnitude at most $2/C^2+2/C$. Taking $C$ sufficiently large keeps its real part positive and comparable to one, so the Fourier transform has magnitude at least a constant times the cap mass, which is at least $\delta /(\pi C)$. No upper bound on the cutoff is required for this lower bound.