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


2010 Mathematics Subject Classification:

11A55, 11B05, 11Y65.

Keywords:

continued fractions, Hall's theorem.

Abstract:

Let be integers, and denote by the set of all irrationals from whose partial quotients of the continued fraction expansion satisfy , . It is proved that every real number can be expressed as a sum of an integer and irrationals from , where a lower bound for is explicitly given. This includes Hall's theorem, and, e.g., the case where , .
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.

Download paper:


Vol. 9 (17), 2014
