On a conjecture of Mahler

Dumir, V. C. ; Hans-Gill, R. J. (1977) On a conjecture of Mahler Bulletin of the Australian Mathematical Society, 16 (01). pp. 125-129. ISSN 1755-1633

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Related URL: http://dx.doi.org/10.1017/S0004972700023078


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.

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