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International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 609-612

Short proofs of theorems of Lekkerkerker and Ballieu

Eberhardstr. 14, 79 Ulm/Donau, Germany

Received 16 October 1981

Copyright © 1982 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For any irrational number ξ let A(ξ) denote the set of all accumulation points of {z:z=q(qξp),p,q{0},pandqrelatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The set A(ξ) is discrete and does not contain zero if and only if ξ is a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.