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Abstract
We develop the inhomogeneous counterpart to some key aspects of the story of the DuffinSchaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sansmonotonicity 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 nonnegative 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 DuffinSchaeffer Conjecture and DuffinSchaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's ZeroOne Law (1961) for inhomogeneous approximation by reduced fractions.
Original language  English 

Pages (fromto)  633654 
Journal  International Journal of Number Theory 
Volume  13 
Issue number  3 
Early online date  29 Sep 2016 
DOIs  
Publication status  Published  Apr 2017 
Bibliographical note
19 pages; v2: changed Erd{\"o}s to Erd\H{o}s throughout. This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy. Further copying may not be permitted; contact the publisher for detailsKeywords
 math.NT
 math.DS
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research