Short proofs of theorems of Lekkerkerker and Ballieu

[摘要] For any irrational numberξletA(ξ)denote the set of all accumulation points of{z:z=q(qξ−p),   p∈ℤ,   q∈ℤ−{0},   p   and   q   relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The setA(ξ)is discrete and does not contain zero if and only ifξis a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.

[学科分类] 数学（综合）