Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

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Abstract

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin-Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Schmidt's inhomogeneous (1964) version of Khintchine's Theorem (1924). For example, given any real sequence $\{y_i \}$, we build a divergent series of non-negative reals $\psi(n)$ such that for any $y\in\{y_i\}$, almost no real number is inhomogeneously $\psi$-approximable with inhomogeneous parameter $y$. Furthermore, given any second sequence $\{z_i\}$ not intersecting the rational span of $\{1,y_i\}$, we can ensure that almost every real number is inhomogeneously $\psi$-approximable with any inhomogeneous parameter $z\in\{z_i\}$. (This extension depends on a dynamical version of Erdos' Covering Systems Conjecture (1950).) Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin-Schaeffer Conjecture and Duffin-Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.
Original languageEnglish
Pages (from-to)633-654
JournalInternational Journal of Number Theory
Volume13
Issue number3
Early online date29 Sep 2016
DOIs
Publication statusPublished - Apr 2017

Bibliographical note

19 pages; v2: changed Erd{\"o}s to Erd\H{o}s throughout. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

Keywords

  • math.NT
  • math.DS

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