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Licensed Unlicensed Requires Authentication Published by De Gruyter March 19, 2010

On Normal Numbers and Powers of Algebraic Numbers

  • Hajime Kaneko
From the journal

Abstract

Let α > 1 be an algebraic number and ξ > 0. Denote the fractional parts of ξαn by {ξαn}. In this paper, we estimate a lower bound for the number λN (α, ξ) of integers n with 0 ≤ n < N and

Our results show, for example, the following: Let α be an algebraic integer with Mahler measure M(α) and ξ > 0 an algebraic number with ξ ∉ ℚ (α). Put [ℚ (α, ξ) : ℚ (α)] = D. Then there exists an absolute constant c satisfying

for all large N.

Received: 2009-07-14
Accepted: 2009-11-17
Published Online: 2010-03-19
Published in Print: 2010-March

© de Gruyter 2010

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