Vol. 20, No. 1, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 298: 1  2
Vol. 297: 1  2
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Other MSP Journals
Some metrical theorems in number theory

Walter Philipp

Vol. 20 (1967), No. 1, 109–127
Abstract

In this paper some metrical theorems on Diophantine approximation, continued fractions and 𝜃-adic expansions are proved.

In the first part some of the common properties of the following transformations from the unit interval onto itself are investigated. Denote by {α} the fractional part of x,

A. T;α →{}a > 1 integer which describes the expansion of α in the scale a

B. T;α →{1
α} which describes the continued fractions

C. T : α →{𝜃α}𝜃 > 1 noninteger which describes the expansion of α as a 𝜃-adic fraction.

The main theorem of the first part (Theorem 2) gives an estimate of the number of solutions of the system of inequalities

Tkα ∈ Ik  1 ≦ k ≦ n

where n is an integer, T is any of these three transformations and (Ik) is an arbitrary sequence of intervals contained in the unit interval.

Mathematical Subject Classification
Primary: 10.33
Secondary: 10.55
Milestones
Received: 15 March 1965
Published: 1 January 1967
Authors
Walter Philipp
http://www.stat.uiuc.edu/people/faculty/philipp.html