Abstract

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

Highlights

  • [44] Mockenhaupt and Tao introduced a variant of the classical Fourier restriction problem in the setting of finite fields

  • The existence of an effective Knapp example in the discrete setting suggests that many of the underlying geometric features of the euclidean Fourier restriction problem should admit some analogue over Z/NZ. This is explored in detail in the current section; in particular, it is shown that there exists a notion of wave packet decomposition over Z/NZ and this leads one to consider certain discrete variants of the Kakeya conjecture

  • It is natural to ask whether the ε-loss in N is necessary in Conjecture 4.2: that is, whether there exists a dimensional constant cn > 0 such that N−n|K| ≥ cn holds for all Kakeya sets K ⊆ [Z/NZ]n

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Summary

Introduction

In [44] Mockenhaupt and Tao introduced a variant of the classical (euclidean) Fourier restriction problem in the setting of finite fields. As the estimate (1) indicates, here the dual group G is equipped with normalised counting measure whereas counting measure is used for the Haar measure on the original group G These choices for Haar measure define the corresponding Lebesgue r norms on these groups and the Fourier transform of any F : Gn → C by F (ξ ) = ∑x∈G F(x)ξ (−x) where ξ denotes a character in the dual group. An investigation of this problem was initiated in [44] in the case where G is a finite-dimensional vector space over a finite field. The Fourier restriction problem is precisely formulated in the setting of [Z/NZ]n; see (4)

The basic setup
Tools and considerations arising from restriction theory
Discrete formulations of the Kakeya conjectures
Sharpness of the Kakeya conjecture
Remaining estimates and identities
Analysis over the p-adic field
Restriction and Kakeya over the p-adics
Correspondence for directions
A correspondence principle for functions
Restriction and Kakeya over local fields
An abstract restriction theorem
Preliminary discussion
Necessary conditions
Sufficient conditions
A Counting solutions to linear systems of congruences

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