The Fourier restriction and Kakeya problems over rings of integers modulo N

Abstract

The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo N for general N and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

“…While the additive combinatorics techniques that preceded the polynomial method [Bou99,KT02] work over any abelian ring, they currently only lead to bounds of the form |S| |R| αn with the α < 0.6. Another, more recent, work to study Kakeya sets (and related operators) over finite rings is [HW18] in which a connection between bounds for Kakeya sets over the rings Z/p k Z and the Minkowski dimension of p-adic Kakeya sets is established. For the two dimensional case, when R = F q [x]/ x k or R = Z/p k Z, Dummit and Hablicsek showed a (tight) bound of |S| |R| 2 /2k in [DH13a].…”

“…Similar to the Euclidean setting, Kakeya sets with Haar measure 0 can be constructed for the ring of p-adic integers and the power series ring F q [[x]]. The constructions can be found in [DH13a,Fra16,Car18,HW18]. As in the Euclidean setting, we want to bound the Minkowski dimension of Kakeya sets for these rings which is connected to the size of Kakeya sets in Z/p k Z and F q [x]/ x k .…”

“…The Kakeya conjecture over the rings R = Z/N Z was stated in [HW18] as follows. As seen above, for finite fields the loss of in the exponent is not necessary.…”

“…Hence, when all the prime factors p i of N are sufficiently large (also as a function of r), we see that the main term in the construction is off by at most a factor of 2 r = N o(1) from the lower bound of Theorem 1.3. Notice that, while the upper bound is obtained from a product of Kakeya 1 In both [EOT10,HW18], where the emphasis was on rings such as Z/p k Z or F q [x]/x k , the definition of projective space over a ring R require the direction b to have at least one invertible coordinate. This definition leads to the same notion of projective space as ours for the rings Z/p k Z or F q [x]/x k .…”

The Kakeya conjecture on local fields of positive characteristic

“…Fraser [Fra16] constructed Kakeya sets of measure zero in L n , where L is any non-archimedean local field with finite residue field. Hickman and Wright [HW18] studied the Kakeya conjecture on Z/N Z in relation with the discrete restriction conjecture, and they gave a simpler construction of a Kakeya set in Z n p of measure zero. Caruso [Car18] has shown that almost all Kakeya needle sets have measure zero in the non-archimedean case.…”