Vol. 20, No. 1, 1967

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Some metrical theorems in number theory

Walter Philipp

Vol. 20 (1967), No. 1, 109–127

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
Received: 15 March 1965
Published: 1 January 1967
Walter Philipp