Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension

TL;DR

This paper demonstrates the existence of zero Fourier dimension sets where the Fourier transform can be restricted on a non-trivial range, extending restriction theorems to more singular sets using novel measure constructions.

Contribution

It introduces a new class of zero Fourier dimension sets allowing Fourier restriction, expanding the scope of restriction theorems beyond positive Fourier dimension cases.

Findings

Existence of zero Fourier dimension sets with non-trivial Fourier restriction

Construction of measures with polylogarithmic decay and full Hausdorff dimension

Extension of the Stein-Tomas-Mockenhaupt Restriction Theorem to new settings

Abstract

In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This builds upon Mockenhaupt's Restriction Theorem; while this theorem could only be applied to sets of positive Fourier dimension, we show that the existence of a measure with polylogarithmic Fourier decay combined with full Hausdorff dimension 1 on the real line is enough to guarantee restriction. In order to achieve this, we combine two different tools: a modification of a construction from a recent work of Li and Liu to produce a set with specific Hausdorff and Fourier dimensions, and a generalization of the Stein-Tomas-Mockenhaupt Restriction Theorem.