= Local fourth-moment restriction estimate for the circle
{title2=$\|\widehat{f\sigma}\|_{L^4(B_R)}\lesssim(\log R)^{1/4}\|f\|_\infty$}
On the <unit circle>, the arclength-measure transform satisfies $|\widehat\sigma(x)|\lesssim(1+|x|)^{-1/2}$. This follows by removing angular neighborhoods of width $|x|^{-1/2}$ around the two stationary points and integrating by parts on the remainder. <Gaussian positivity for even extension moments> bounds the weighted fourth moment for any bounded $f$ by that for $f=1$. Radial integration of $(1+r)^{-2}e^{-\pi r^2/R^2}$ with area element $r\,dr$ costs only $\log R$, proving the displayed local estimate. Modulated disjoint caps and random signs convert it into the <planar Kakeya neighborhood lower bound>.
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