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On the Powers of Some Transcendental Numbers

Published online by Cambridge University Press:  17 April 2009

Artūras Dubickas
Artūras Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania, e-mail: arturas.dubickas@mif.vu.lt
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We construct a transcendental number α whose powers αn!, n = 1, 2, 3,…, modulo 1 are everywhere dense in the interval [0, 1]. Similarly, for any sequence of positive numbers δ = (δn)∞n=1, we find a transcendental number α = α(δ) such that the inequality {αn} < δn holds for infinitely many n ∈ N, no matter how fast the sequence δ converges to zero. Finally, for any sequence of real numbers (rn)∞n=1 and any sequence of positive numbers (δn)∞n=1, we construct an increasing sequence of positive integers (qn)∞n=1 and a number α > 1 such that ‖αqn – τn‖ < δn for each n ≥ 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 433 - 440
Copyright
Copyright © Australian Mathematical Society 2007

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