Compact-set Fourier amplification lemma ($\Vert q{\Vert }_{\infty }\le \epsilon ,|S_{L}q|>M\mathrm{on}K$)

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, 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.

Let finitely many arcs have centers ${\theta }_j$, half-lengths $0<{\delta }_j\le 1$ and $D={\sum }_j{\delta }_j$. Each summand has positive real part on the closed disk, and $\Phi (0)=1$. On the $j$th arc its contribution has real part at least $1/(4D)$. Thus the analytic logarithm $A=\log \Phi$ obeys $A(0)=0$, $|\mathrm{Im}A|<\pi /2$, and $\mathrm{Re}A\ge \log (1/(4D))$ throughout the arc union. Approximate $A$ by an analytic polynomial and modulate its bounded imaginary part; a partial Fourier sum isolates one half of the polynomial, producing a large value from a bounded function.