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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the digits of squares and the distribution of quadratic subsequences of digital sequences


Authors: Roswitha Hofer, Gerhard Larcher and Heidrun Zellinger
Journal: Proc. Amer. Math. Soc. 141 (2013), 1551-1565
MSC (2010): Primary 11K06, 11K31, 11K38
Published electronically: October 31, 2012
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Abstract: Let $ q$ be a prime and $ \gamma =(\gamma _0,\,\gamma _1,\,\gamma _2,\,\dots )$ with $ \gamma _i\in \{0,\,\dots ,\,q-1\}$ and $ \gamma _i=0$ for $ i\geq i_0$ be a weight sequence.

We study $ \lim _{N\rightarrow \infty }{\char93 \left \{0\leq n< N\vert s_{q,\gamma }(n^2)\equiv d\,(\mod q)\right \}}/N$ with $ s_{q,\gamma }(n^2)$ the weighted sum of digits of $ n^2$ in base $ q$ and we use the results to classify the digital sequences $ \left (\boldsymbol {x}_n\right )_{n\geq 0}$ in the sense of Niederreiter, generated by matrices with finite rows, for which $ \left (\boldsymbol {x}_{n^2}\right )_{n\geq 0}$ is uniformly distributed. Finally we derive an upper bound for the star discrepancy of these uniformly distributed subsequences.


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Additional Information

Roswitha Hofer
Affiliation: Institute of Financial Mathematics, University of Linz, Altenbergerstr. 69, 4040 Linz, Austria
Email: roswitha.hofer@jku.at

Gerhard Larcher
Affiliation: Institute of Financial Mathematics, University of Linz, Altenbergerstr. 69, 4040 Linz, Austria
Email: gerhard.larcher@jku.at

Heidrun Zellinger
Affiliation: Institute of Financial Mathematics, University of Linz, Altenbergerstr. 69, 4040 Linz, Austria
Email: heidrun.zellinger@jku.at

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11448-0
PII: S 0002-9939(2012)11448-0
Keywords: Uniform distribution, discrepancy, digital $(\mathbf{T},s)$-sequences, quadratic subsequences, digits of squares
Received by editor(s): April 26, 2011
Received by editor(s) in revised form: September 5, 2011
Published electronically: October 31, 2012
Additional Notes: The first author was supported by the Austrian Science Fund under Project Nr. P21943.
The second author was supported by the Austrian Science Fund under Projects Nr. P21196 and P21943.
The third author is a recipient of a DOC-fForte Grant of the Austrian Academy of Sciences at the Institute of Financial Mathematics at the University of Linz (Austria)
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society