English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Metric Diophantine approximation in its classical form is the study of howwell almost all real numbers can be approximated by rationals. There is a longhistory of results which give partial answers to this problem, but there arestill questions which remain unknown. The Duffin-Schaeffer Conjecture is anattempt to answer all of these questions in full, and it has withstood morethan fifty years of mathematical investigation. In this paper we establish astrong connection between the Duffin-Schaeffer Conjecture and its p-adicanalogue. Our main theorems are transfer principles which allow us to go backand forth between these two problems. We prove that if the variance method fromprobability theory can be used to solve the p-adic Duffin-Schaeffer Conjecturefor even one prime p, then almost the entire classical Duffin-SchaefferConjecture would follow. Conversely if the variance method can be used to provethe classical conjecture then the p-adic conjecture is true for all primes.Furthermore we are able to unconditionally and completely establish the higherdimensional analogue of this conjecture in which we allow simultaneousapproximation in any finite number and combination of real and p-adic fields,as long as the total number of fields involved is greater than one. Finally byusing a mass transference principle for Hausdorff measures we are able toextend all of our results to their corresponding analogues with Haar measuresreplaced by the Hausdorff measures associated with arbitrary dimensionfunctions.

The metric theory of p-adic approximation