Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

TL;DR

This paper establishes a Khintchine-type theorem for multiplicative Diophantine approximation on lines with restricted denominators, determining convergence conditions for measure and dimension of certain approximation sets.

Contribution

It extends multiplicative Diophantine approximation results to arbitrary non-degenerate lines with restricted denominators, providing convergence criteria and Hausdorff dimension bounds.

Findings

Sets satisfying the approximation condition have zero Hausdorff s-measure under certain convergence.

Derived an upper bound for Hausdorff dimension in the inhomogeneous case.

Generalized Khintchine results to a broader class of Diophantine approximation problems.

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 < \psi(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.