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arXiv · 2606.27034

Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

Prasuna Bandi
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Abstract

Let $C$ be the middle-third Cantor set and let $\mu$ be the natural Cantor probability measure. Let \[ \gamma=\frac{\log2}{\log3}. \] The two main results of this paper are \[ \mu\{x\in C:\|2^n x\| 2-\gamma. \] and \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=1 \qquad \text{ for } \tau<\frac{1-\gamma}{2}. \] These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.

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Prasuna Bandi. 2026-06-25. Averaged Fourier Estimates and Dyadic Approximation on the Cantor set. https://arxiv.org/abs/2606.27034

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