On linear normal lattices configurations

Mordechay B. Levin[1]; Meir Smorodinsky[2]

  • [1] Department of Mathematics Bar-Ilan University 52900, Ramat-Gan, Israel
  • [2] School of Mathematical Sciences Tel Aviv University 69978, Tel-Aviv, Israel

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 3, page 825-858
  • ISSN: 1246-7405

Abstract

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In this paper we extend Champernowne’s construction of normal numbers in base b to the ℤ d case and obtain an explicit construction of the generic point of the ℤ d shift transformation of the set { 0 , 1 , . . . , b - 1 } ℤ d . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base b .

How to cite

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Levin, Mordechay B., and Smorodinsky, Meir. "On linear normal lattices configurations." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 825-858. <http://eudml.org/doc/249467>.

@article{Levin2005,
abstract = {In this paper we extend Champernowne’s construction of normal numbers in base $b$ to the $\mathbb\{Z\}^d$ case and obtain an explicit construction of the generic point of the $\mathbb\{Z\}^d$ shift transformation of the set $\lbrace 0,1,...,b-1 \rbrace ^\{\mathbb\{Z\}^d\}$. We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base $b$ .},
affiliation = {Department of Mathematics Bar-Ilan University 52900, Ramat-Gan, Israel; School of Mathematical Sciences Tel Aviv University 69978, Tel-Aviv, Israel},
author = {Levin, Mordechay B., Smorodinsky, Meir},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {normal lattice configurations; normal numbers},
language = {eng},
number = {3},
pages = {825-858},
publisher = {Université Bordeaux 1},
title = {On linear normal lattices configurations},
url = {http://eudml.org/doc/249467},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Levin, Mordechay B.
AU - Smorodinsky, Meir
TI - On linear normal lattices configurations
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 825
EP - 858
AB - In this paper we extend Champernowne’s construction of normal numbers in base $b$ to the $\mathbb{Z}^d$ case and obtain an explicit construction of the generic point of the $\mathbb{Z}^d$ shift transformation of the set $\lbrace 0,1,...,b-1 \rbrace ^{\mathbb{Z}^d}$. We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base $b$ .
LA - eng
KW - normal lattice configurations; normal numbers
UR - http://eudml.org/doc/249467
ER -

References

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  2. D. J. Champernowne, The construction of decimals normal in the scale of ten. J. London Math. Soc. 8 (1933), 254–260. Zbl0007.33701
  3. J. Cigler, Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math. 64 (1960), 201–225. Zbl0111.25301MR121358
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  5. T. Kamae, Subsequences of normal sequences. Israel J. Math. 16 (1973), 121–149. Zbl0272.28012MR338321
  6. N. M. Korobov, Exponential Sums and their Applications. Kluwer Academic Publishers, Dordrecht, 1992. Zbl0754.11022MR1162539
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  8. P. Kirschenhofer, R.F. Tichy, On uniform distribution of double sequences. Manuscripta Math. 35 (1981), 195–207. Zbl0478.10036MR627933
  9. M. B. Levin, On normal lattice configurations and simultaneously normal numbers. J. Théor. Nombres Bordeaux 13 (2001), 483–527. Zbl0999.11039MR1879670
  10. M. B. Levin, Discrepancy estimate of completely uniform distributed double sequences. In preparation. 
  11. M. B. Levin, M. Smorodinsky, A ℤ d generalisation of the Davenport–Erdös construction of normal numbers. Colloq. Math. 84/85 (2000), 431–441. Zbl1014.11046MR1784206
  12. M. B. Levin, M. Smorodinsky, Explicit construction of normal lattice configurations. Colloq. Math. 102 (2005), 33–47. Zbl1080.11057MR2150267
  13. M. B. Levin, M. Smorodinsky, On polynomial normal lattice configurations. Monatsh. Math. (2005) Zbl1126.11037MR2216557
  14. A. G. Postnikov, Arithmetic modeling of random processes. Proc. Steklov. Inst. Math. 57 (1960), 84 pp. Zbl0106.12101MR148639
  15. M. Smorodinsky, B. Weiss, Normal sequences for Markov shifts and intrinsically ergodic subshifts. Israel J. Math. 59 (1987), 225–233. Zbl0643.10041MR920084
  16. B. Weiss, Normal sequences as collectives. Proc. Symp. on Topological Dynamics and Ergodic Theory, Univ. of Kentucky, 1971. 

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