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 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.