Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin--Schaeffer conjecture for a class of non-monotonic approximation functions.

Chapter 1. Introduction 1

1.1. Main results 5

1.2. Key ideas and further results 7

1.3. Open problems 11

1.4. Organization and notation 13

      Acknowledgements 14

Chapter 2. Preliminaries 15

2.1. Continued fractions 15

2.2. Ostrowski expansions 17

2.3. Bohr sets 19

2.4. Measure theory 21

2.5. Real analysis 21

2.6. Geometry of numbers 23

2.7. Primes and sieves 25

Chapter 3. A fully inhomogeneous version of Gallagher's theorem 27

3.1. Notation and reduction steps 27

3.2. Divergence of the series 29

3.3. Overlap estimates, localized Bohr sets, and the small-GCD regime 35

3.4. Large GCDs 40

3.5. A convergence statement 45

Chapter 4. Liouville fibres 49

4.1. A special case 49

4.2. Diophantine second shift 51

4.3. Liouville second shift 55

4.4. Rational second shift 59

Chapter 5. Obstructions on Liouville fibres 61

Appendix A. Pathology 67

Bibliography 71

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