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

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 Szüsz's inhomogeneous (1958) version of Khintchine's Theorem (1924). For example, given any real sequence , we build a divergent series of non-negative reals such that for any , almost no real number is inhomogeneously -approximable with inhomogeneous parameter . Furthermore, given any second sequence not intersecting the rational span of , and assuming a dynamical version of Erdős' Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously -approximable with any inhomogeneous parameter . 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. [ABSTRACT FROM AUTHOR]