Digital expansion of exponential sequences

Michael Fuchs

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 477-487
  • ISSN: 1246-7405

Abstract

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We consider the -ary digital expansion of the first terms of an exponential sequence . Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first digits, where is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo plays an important role.

How to cite

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Fuchs, Michael. "Digital expansion of exponential sequences." Journal de théorie des nombres de Bordeaux 14.2 (2002): 477-487. <http://eudml.org/doc/248923>.

@article{Fuchs2002,
abstract = {We consider the $q$-ary digital expansion of the first $N$ terms of an exponential sequence $a^n$. Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last $c \log N$ digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first $(\log N)^\{\frac\{3\}\{2\}-\epsilon \}$ digits, where $\epsilon $ is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo $1$ plays an important role.},
author = {Fuchs, Michael},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {digital expansion; exponential sequence; subblock occurrence; sum-of-digits function},
language = {eng},
number = {2},
pages = {477-487},
publisher = {Université Bordeaux I},
title = {Digital expansion of exponential sequences},
url = {http://eudml.org/doc/248923},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Fuchs, Michael
TI - Digital expansion of exponential sequences
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 477
EP - 487
AB - We consider the $q$-ary digital expansion of the first $N$ terms of an exponential sequence $a^n$. Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last $c \log N$ digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first $(\log N)^{\frac{3}{2}-\epsilon }$ digits, where $\epsilon $ is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo $1$ plays an important role.
LA - eng
KW - digital expansion; exponential sequence; subblock occurrence; sum-of-digits function
UR - http://eudml.org/doc/248923
ER -

References

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