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230 KB, PDF document

231 KB, PDF document

Journal | International Journal of Number Theory |
---|---|

Date | In preparation - 18 Aug 2015 |

Date | Accepted/In press - 16 Mar 2016 |

Date | E-pub ahead of print - 29 Sep 2016 |

Date | Published (current) - Apr 2017 |

Issue number | 3 |

Volume | 13 |

Pages (from-to) | 633-654 |

Early online date | 29/09/16 |

Original language | English |

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.

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

- math.NT, math.DS

## Programme Grant-New Frameworks in metric Number Theory

Project: Research project (funded) › Research

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