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arXiv · 2409.03331

Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers

Bo Tan·Qing-Long Zhou
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Abstract

Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.

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Bo Tan, Qing-Long Zhou. 2024-09-05. Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers. https://arxiv.org/abs/2409.03331

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