New Kakeya estimates using the polynomial Wolff axioms

Abstract

We obtain new bounds for the Kakeya maximal conjecture in most dimensions n < 100, as well as improved bounds for the Kakeya set conjecture when n = 7 or 9. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when n = 5 or n 7.

New Kakeya estimates using Gromov's algebraic lemma

“…Lemma 1.4 is standard. A variant of the lemma was proved by Bourgain and Guth [5] in the context of the restriction problem, and a version similar to the one stated here can be found in [14]. Combining Theorem 1.3 and Lemma 1.4, we obtain the following bounds on the Kakeya maximal function.…”

“…Corollary 1.8 was used by Guth and the author in [12] 2 and by Katz and the author in [18] to obtain improved bounds on the Kakeya maximal function in R 4 . It was used by Hickman and Rogers [13,14] to obtain improved Kakeya bounds for certain dimensions n ≥ 5, and to obtain improved restriction estimates in dimension ≥ 13, as well as dimension 4, 5, 7, 9, and 11.…”

“…Note that when k < n, the value of d from (1.4) is larger than k. Theorem 1.3 generalizes a previous result of Guth and the author [12], which proved 1 Theorem 1.3 in the special case d = 3, n = 4. The techniques in [12] naturally extend to the case k = n−1, and they can also be used to prove weaker variants of Theorem 1.3 for general n and k (Hickman and Rogers [14] employed a similar strategy to prove certain k-broad estimates in R n ). However, several additional ideas are needed when k < n − 1.…”

“…In high dimensions, the previous best-known bound on the Kakeya maximal function was d(n) = (4n + 3)/7, due to Katz and Tao[16]. In certain intermediate dimensions 5 ≤ n ≤ 100 the previous best-known bound was due to Hickman and Rogers[14].Remark 1.6. Recently, the author became aware that Hickman, Rogers, and Zhang have concurrently and independently proved Theorem 1.5.…”