= Compact-set Fourier amplification lemma
{title2=$\|q\|_\infty\leq\varepsilon,\quad |S_Lq|>M\ {\rm on}\ K$}
For a compact circle set with normalized <Lebesgue measure> at most $\exp(-8\pi M/\varepsilon)$, where $M\geq1$ and $0<\varepsilon\leq1$, the displayed bounds can be realized by a <trigonometric polynomial> with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part <Schwarz integral on the unit disk> of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large <Fourier partial sums> drives the <Kahane-Katznelson divergence theorem>.
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