Short proofs of theorems of Lekkerkerker and Ballieu

Abstract

For any irrational number ξ let A(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 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.