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Bernoulli polynomials and $(n alpha)$-sequences

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Author(s): Luís Roçadas

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 3;
Issue: 1;
Start page: 127;
Date: 2008;
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Keywords: Bernoulli polynomial | fractional part | continued fraction

ABSTRACT
Let $alphain(0,1)$ be an irrational with continued fractionexpansion $alpha=[0;a_1,...]$ and convergents$frac{p_n}{q_n}, n = 0,1, dots .$ Given a positive integer $N$ thereexists a unique digit expansion, $N=sum_{i=0}^m b_i q_i$,where the digits $b_i$ are non-negative integers satisfying theconditions $b_02$. The formula for $u=2$ allows us to compute$sum_{n =1}^N B_2(nalpha)$ in $O((log N)^3)$ steps. Finally we determineall of this $alpha's$ for which this sum is bounded.
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