Improved bounds for the Kakeya maximal conjecture in higher dimensions

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to context of Kakeya problem. By writing out induction as a recursive algorithm, additional multiscale geometric information is made available. To take advantage this, we prove that direction-separated tubes satisfy version Wolff axioms. Altogether, this yields improved bounds maximal conjecture in~$\Bbb{R}^n$ with $n=5$ or $n\ge 7$ and set an infinite sequence dimensions.