## Large sets without Fourier restriction theorems

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- by Constantin Bilz PDF
- Trans. Amer. Math. Soc.
**375**(2022), 6983-7000 Request permission

## Abstract:

We construct a function that lies in $L^p(\mathbb {R}^d)$ for every $p \in (1,\infty ]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.## References

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## Additional Information

**Constantin Bilz**- Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England
- Email: c.bilz@pgr.bham.ac.uk
- Received by editor(s): October 25, 2020
- Received by editor(s) in revised form: December 31, 2021
- Published electronically: August 11, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 6983-7000 - MSC (2020): Primary 42B10; Secondary 28A80
- DOI: https://doi.org/10.1090/tran/8714