Some metrical theorems in number theory

Walter Philipp

Vol. 20 (1967), No. 1, 109–127

DOI: 10.2140/pjm.1967.20.109

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 →{aα}a > 1 integer which describes the expansion of α in the scale a

B. T;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

where n is an integer, T is any of these three transformations and (I_k) 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