| Summary: | The central aim of this thesis is to prove various Diophantine approximation results which quantify cases of the weak approximation theorem. We take finitely many different completions of a global field , take their direct product, and approximate elements of this space by elements of the global field. We prove analogues of Gallagher's zero-one law, and of the Duffin-Schaeffer theorem, in two setups of this type: the direct product of finitely many completions of Q (always including R), and the direct product of all of the Archimedean completions of a general number field . The second result in particular forms a significant improvement on existing results, which were only proven for imaginary quadratic fields.
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