= Circle cap Fourier lower bound
{title2=$|\widehat{\psi\,d\sigma}|\gtrsim\delta\text{ on a }\delta^{-2}\times\delta^{-1}\text{ rectangle}$}
For normalized circle measure and a nonnegative cutoff near $(1,0)$, equal to one on $|\theta|\le\delta/C$ and supported on $|\theta|\le2\delta/C$, 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.
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