This article is cited in 1 paper

On the Mean Value of the Measure of Irrationality of Real Numbers

D. O. Shatskov

Astrakhan State University

Abstract: This paper deals with the asymptotic behavior of the integral
$$ I_\alpha(t)=\int_1^t \psi_\alpha(\xi)\,d\xi, \qquad\text{where}\quad \psi_\alpha(t)=\min_{1\le q\le t}\|q\alpha\| $$
(here the minimum is taken over integers $q$ and $\|\,\cdot\,\|$ denotes the distance to the nearest integer).

Keywords: real number, measure of irrationality, continued fraction, convergent, Lebesgue measure, Gauss transformation, ergodic transformation.

UDC: 517

Received: 07.04.2012
Revised: 01.03.2015

DOI: 10.4213/mzm9374

 English version: Mathematical Notes, 2015, 98:2, 301–315

Bibliographic databases:

• 
• 
• 
•