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A new type of deterministic Salem sets and its spectrality
Abstract
For all 0< s ≤ 1, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to s. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis {e2πi λx: λ∈ Λ} for its L2 space. This gives the first examples of singular Salem spectral measures in ℝ1.
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BibTeX
@misc{lai2026type,
title = {{A new type of deterministic Salem sets and its spectrality}},
author = {Chun-Kit Lai and Ruxi Shi and Yu-Hao Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22772},
eprint = {2609.22772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22772},
}
RIS
TY - UNPB
TI - A new type of deterministic Salem sets and its spectrality
AU - Chun-Kit Lai
AU - Ruxi Shi
AU - Yu-Hao Xie
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22772
UR - https://arxiv.org/abs/2609.22772
AB - For all 0< s ≤ 1, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to s. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis {e^2πi λx: λ∈ Λ} for its L^2 space. This gives the first examples of singular Salem spectral measures in ℝ^1.
ER -