From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE (Q2756758)

The author reviews certain connections between the Kakeya problem and several problems in the theory of oscillatory integrals and PDEs. For \(n\geq 2\), a Besicovitch set is a subset of \(R^{n}\) that contains a unit line segment in every dimension. Such sets can have Lebesgue measure zero. The Kakeya conjecture, which holds in \(R^{2}\) but remains open in higher dimensions, asserts essentially that the volume of a \(\delta \) neighborhood of a Besicovitch set in \(\mathbb{R}^{n}\) is bounded below by \(C_{n,\varepsilon }\delta ^{\varepsilon }\) for any \(\varepsilon >0\) and \(\delta \) small enough. In discrete form, the conjecture asserts that if \(\Omega \) is a maximally \(\delta \)-separated subset of the sphere and if one considers a \(\delta \times 1\) tube \(T_{\omega }\) in the unit ball along the direction \(\omega \in \Omega \) then \(|\cup _{\omega \in \Omega }T_{\omega }|\) must have a certain type of logarithmic -- really iterated logarithmic -- growth in \(\delta \). Such collections of tubes become useful in studying oscillatory integrals and PDEs since, as C. Fefferman showed in his work on the disk multiplier, the tubes can fit wavepackets whose interactions can cause the failure of boundedness of fundamental operators such as the disk multiplier \(S_{1}f(x)=\int_{|\xi |<1}\widehat{f}(\xi)e^{2\pi ix\xi } d\xi \) where \(\widehat{f} \) denotes the Fourier transform of \(f\). The author briefly outlines this and several other constructions for related problems, namely Bochner-Riesz summability of Fourier series, as well as C. Sogge's local smoothing conjecture \(\|u\|_{L^{p}([1,2]\times \mathbb{R}^{n})}\leq C_{p,\varepsilon }\|(1+\sqrt{-\Delta })f\|_{L^{p}(\mathbb{R}^{n})}\) whenever \(\varepsilon >\max (0,n(\frac{1}{2}-\frac{1}{p})-\frac{1}{2})\) and \( 2\leq p\leq \infty \). The author also alludes to combinatorial methods that can be used in proving finite field analogues of the conjecture itself, as he has done in more recent work.