Kakeya conjecture

There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the  result established by Tomas Wolff in 1995 is almost the best result in $ R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate. For $ f\in L_{loc}^1(R^d)$,for $ 0<\delta<1$: $ f_{\delta}^*:P^{d-1}\longrightarrow R$.$ f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$. $ T$ is varise in all cylinders with length 1.radius $ \delta$.axis in…

Kakeya conjecture in $latex R^3$ is very subtle.in fact wolff stay the best(but not very difficult to get,just use the structure so-called hairbrush)result $latex \frac{5}{2}$ until the result of Katz and Tao $latex \frac{5}{2}+\epsilon$.Where $latex \epsilon$ is a constant independent with kakeya set.and in the article of Tao,they proved $latex \epsilon>\frac{1}{10^{10}}$. Two-dimensional case first we overview the case of dimension 2,these is the only case that is proved.and the key point i…

Last year I read a nice blog articles Recent progress on the Kakeya conjecture and have several questions with this article. follows the proof strategy called Multiscale analysis,although we can use the estimate with large $ \delta_1$ to get estimate with small $ \delta_2$,(may be loss some $ \delta^c$ in the inequality in this way),but the main difficult is we should proof the new tubes with scales $ \delta_2$ is contains in the the olders.as soon as we proof this ,to obtain a lower b…