ON A GENERALIZATION OF HALL'S THEOREM FOR SUMS OF CONTINUED FRACTIONS

Author:

Carsten~Elsner

2010 Mathematics Subject Classification:

11A55, 11B05, 11Y65.

Keywords:

continued fractions, Hall's theorem.

Abstract:

Let \(1\leq M<N \) be integers, and denote by \(CF(M,N)\) the set of all irrationals from \([0,1] \) whose partial quotients \(a_{\nu} <span style="font-size: 1em; line-height: 1.3em;">\) of the continued fraction expansion satisfy \(M\leq a_{\nu} \leq N\), \(\nu \geq 1\). It is proved that every real number can be expressed as a sum of an integer and \(m\) irrationals from \(CF(M,N) \), where a lower bound for </span><span style="font-size: 1em; line-height: 1.3em;">\(m\) is explicitly given. This includes Hall's theorem, and, e.g., the case where \(M=N-1\), \(m=N^2+1 \).

The main theorem is a special case of a more general result of S. Astels from 2000. Our proof is focussed on a generalization of a lemma of Hall concerning the iterated thinning of intervals. For our weaker result one does not need the theory of Cantor sets underlying Astels' approach from 2000.

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Vol. 9 (17), 2014