Projects per year
Abstract
Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the set of real numbers $x$ such that $nx a < \psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\psi)$ is of full Lebesgue measure if there exists an $\epsilon > 0 $ such that $$ \textstyle \sum_{n\in\N}(\frac{\psi(n)}{n})^{1+\epsilon}\varphi (n)=\infty . $$ The DuffinSchaeffer Conjecture is the corresponding statement with $\epsilon = 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(\psi)$ is of full Hausdorff dimension if the above sum with $\epsilon = 0$ diverges; i.e. the dimension analogue of the DuffinSchaeffer Conjecture is true.
Original language  English 

Pages (fromto)  259273 
Number of pages  15 
Journal  Mathematischen Annalen 
Volume  353 
Issue number  2 
DOIs  
Publication status  Published  Jun 2012 
Keywords
 Number Theory
Projects
 2 Finished

Classical metric Diophantine approximation revisited
24/03/08 → 23/07/11
Project: Research project (funded) › Research

Inhomogenous approximation on manifolds
15/02/08 → 14/04/11
Project: Research project (funded) › Research