%0 Journal Article
%T A theorem of Khintchine type
%A Enrico Zoli
%J Uniform Distribution Theory
%D 2008
%I Mathematical Institute of the Slovak Academy of Sciences
%X Let $ \psi : \mathbb{N} \to [0, \infty)$ be an approximation function with $ \sum_{q = 1}^ \infty q \psi (q)$ $ = \infty$ and the property that there exists $ \delta > 0$ such that $\psi(q) \geq \delta \psi(s)$ for all $q \in \mathbb{N}$ and all $s \in {q, q+1, \ldots, 2q\}$. Then the set $$\left\{x \in (0,1) : \left|x - \frac p q \right| < \psi(q) \textrm {forinfinitely many reduced rationals }\frac p q \right\}$$ has Lebesgue measure one.
%K Khintchine theorem
%K Borel--Cantelli lemma
%K approximation function
%K Euler's $ phi$-function
%K Dawson--Sankoff inequality
%K Lebesgue measure
%K Duffin--Schaeffer conjecture
%U http://www.boku.ac.at/MATH/udt/vol03/no1/Zoli08-1.pdf