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Inhomogeneous Diophantine Approximation on $M_0$-sets

Volodymyr Pavlenkov, Evgeniy Zorin

Abstract

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-Järnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators.

Inhomogeneous Diophantine Approximation on $M_0$-sets

Abstract

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on -sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-Järnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators.
Paper Structure (13 sections, 16 theorems, 210 equations)

This paper contains 13 sections, 16 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Khintchine-Szüsz $0$ and $1$ Law and Schmidt generalization
  3. Main result
  4. Applications of main result
  5. Example of sequence $\mathcal{A}$ with polynomial growth and fitting all conditions of Theorem \ref{['mainTHM']}
  6. Khintchine Theorem on the set of Liouville numbers
  7. Hausdorff and Fourier dimensions of sets $W_{\mathcal{A}}(\gamma; \psi)$ and their intersections
  8. Establishing the main result
  9. Choosing $f_n(x)$ and $f_n$ from Lemma \ref{['ebc']}
  10. Estimating the measure of the sets $E_n$ and their intersections
  11. Estimating the sums from measures of $E_n$
  12. Estimating the sums from measures of intersections $E_n \cap E_m$
  13. Applying Lemma \ref{['ebc']} to prove Theorem \ref{['mainTHM']}

Key Result

Proposition 1

Let $x \in \mathbb{I},$$\gamma\in\mathbb{I}$ and let $\psi, \psi_1, \psi_2$ be auxiliary functions, that is $\psi, \psi_1, \psi_2:\mathbb{N}\rightarrow \mathbb{I}$. Let $\mathcal{A} = (q_n)_{n\in \mathbb{N}} \subset \mathbb{N}$ be a sequence of natural numbers. Then, (i)$x \in W_{\mathcal{A}}(\gamma (ii) if $\psi_1(q_n) \le \psi_2(q_n)$ for all $n \in \mathbb{N},$ then and (iii) if for some au

Theorems & Definitions (36)

  • Definition 1
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop1']}
  • Definition 2
  • Remark 1
  • Theorem 1
  • Remark 2
  • Example 1
  • Example 2
  • Theorem 2
  • ...and 26 more