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SUBBLOCK OCCURRENCES IN SIGNED DIGIT REPRESENTATIONS

Published online by Cambridge University Press:  10 September 2003

PETER J. GRABNER ,
CLEMENS HEUBERGER  and
HELMUT PRODINGER
PETER J. GRABNER
Affiliation:
Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria e-mail: peter.grabner@tugraz.at
CLEMENS HEUBERGER
Affiliation:
Institut für Mathematik B, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria e-mail: clemens.heuberger@tugraz.at
HELMUT PRODINGER
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, P. O. Wits 2050 Johannesburg, South Africa e-mail: helmut@staff.ms.wits.ac.za
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Abstract

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Signed digit representations with base $q$ and digits $-\frac q2,\dots,\frac q2$ (and uniqueness being enforced by applying a special rule which decides whether $-q/2$ or $q/2$ should be taken) are considered with respect to counting the occurrences of a given (contiguous) subblock of length $r$. The average number of occurrences amongst the numbers $0,\dots,n-1$ turns out to be const $\cdot\log_qn+\delta(\log_qn)+\smallOh(1)$, with a constant and a periodic function of period one depending on the given subblock; they are explicitly described. Furthermore, we use probabilistic techniques to prove a central limit theorem for the number of occurrences of a given subblock.

Keywords

Primary: 11K16Secondary: 11A6311M41
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 45 , Issue 3 , September 2003 , pp. 427 - 440
Copyright
© 2003 Glasgow Mathematical Journal Trust