On a conjecture of Mahler

Abstract

Let R be the field of real numbers. For a in R, let ||α|| be the distance of a from the nearest integer. The following conjecture of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463-465] is proved. Let m, n be two positive integers n ≥ 2m. Let S be a finite or infinite set of positive integers with the following properties: (Q₁) S contains the integers m, m+1, …, n-m; (Q₂) every element of S satisfies ||θ/n|| ≥ m/n then sup inf||εα|| m/n. αεR eεS.