Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension
Iván Polasek, Ezequiel Rela
TL;DR
The paper solves the restriction problem on a zero-Fourier-dimension set by constructing a fractal $E\subset\mathbb{R}$ with $\dim_H(E)=1$ and $\dim_F(E)=0$ and a measure $\mu$ with polylogarithmic Fourier decay $|\widehat{\mu}(\xi)| \lesssim 1/\log^{r}|\xi|$. The method combines a Li–Liu–type construction with a generalization of the STM restriction theorem to handle the zero Fourier dimension limit, yielding $R_E(p\to 2)$ for $1\le p<1+(r-1)/(1+r+2ar)$. This demonstrates restriction in a regime where Fourier decay is polylogarithmic and the Fourier dimension vanishes, clarifying how growth and dimension functions interact in fractal restriction problems. The results provide explicit deterministic sets and measures that achieve restriction at the limit case and contribute to the broader understanding of fractal harmonic analysis and restriction theory.
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.