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Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

Lucas Tapia

Abstract

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\, \{b_n\}_{n\in\mathbb{N}},\, \{c_n\}_{n\in\mathbb{N}},\, \{d_n\}_{n\in\mathbb{N}},$ we determine convergence conditions under which the set of $x\in [0,1]$ which satisfy $\left\lVert a_n x +c_n\right\rVert \cdot \left\lVert b_n x + d_n \right\rVert < ψ(n) $ for infinitely many $n\in\mathbb{N}$ has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting.

Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

Abstract

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers we determine convergence conditions under which the set of which satisfy for infinitely many has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting.
Paper Structure (8 sections, 13 theorems, 96 equations)

This paper contains 8 sections, 13 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Definition of the Hausdorff measure
  3. Preliminary Estimates
  4. The real case
  5. A covering result and lattice counting
  6. Hausdorff Measure Estimates
  7. The integer case
  8. Convergence Results

Key Result

Theorem 1

Suppose that $\{a_n\}$ and $\{b_n\}$ are restricted to be sequences of positive integers. In particular, it follows that where $\tau$ is the infimum of those $s$ for which integerconv holds.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 1: BV Lemma 3.10
  • Lemma 2
  • Lemma 3
  • Lemma 4: Erdős-Turán Inequality (mo, Theorem 1.1)
  • proof : Proof of Lemma \ref{['key']}
  • Lemma 5
  • proof
  • ...and 11 more