On normal numbers mod 2

Youngho Ahn; Geon Choe

Colloquium Mathematicae (1998)

  • Volume: 76, Issue: 2, page 161-170
  • ISSN: 0010-1354

Abstract

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It is proved that a real-valued function f ( x ) = exp ( π i χ I ( x ) ) , where I is an interval contained in [0,1), is not of the form f ( x ) = q ( 2 x ) ¯ q ( x ) with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.

How to cite

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Ahn, Youngho, and Choe, Geon. "On normal numbers mod $2$." Colloquium Mathematicae 76.2 (1998): 161-170. <http://eudml.org/doc/210556>.

@article{Ahn1998,
abstract = {It is proved that a real-valued function $f(x)=\exp (\pi i \chi _I(x))$, where I is an interval contained in [0,1), is not of the form $f(x)=\overline\{q(2x)\}q(x)$ with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.},
author = {Ahn, Youngho, Choe, Geon},
journal = {Colloquium Mathematicae},
keywords = {coboundary; metric density; normal number; uniform distribution; normal numbers mod 2},
language = {eng},
number = {2},
pages = {161-170},
title = {On normal numbers mod $2$},
url = {http://eudml.org/doc/210556},
volume = {76},
year = {1998},
}

TY - JOUR
AU - Ahn, Youngho
AU - Choe, Geon
TI - On normal numbers mod $2$
JO - Colloquium Mathematicae
PY - 1998
VL - 76
IS - 2
SP - 161
EP - 170
AB - It is proved that a real-valued function $f(x)=\exp (\pi i \chi _I(x))$, where I is an interval contained in [0,1), is not of the form $f(x)=\overline{q(2x)}q(x)$ with |q(x)|=1 a.e. if I has dyadic endpoints. A relation of this result to the uniform distribution mod 2 is also shown.
LA - eng
KW - coboundary; metric density; normal number; uniform distribution; normal numbers mod 2
UR - http://eudml.org/doc/210556
ER -

References

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  1. [1] G. H. Choe, Spectral types of uniform distribution, Proc. Amer. Math. Soc. 120 (1994), 715-722. Zbl0803.11039
  2. [2] G. H. Choe, Ergodicity and irrational rotations, Proc. Roy. Irish Acad. 93A (1993), 193-202. Zbl0807.28009
  3. [3] R. B. Kirk, Sets which split families of measurable sets, Amer. Math. Monthly 79 (1972), 884-886. Zbl0249.28003
  4. [4] K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983. 
  5. [5] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1986. 
  6. [6] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2 , Trans. Amer. Math. Soc. 140 (1969), 1-33. Zbl0201.05601
  7. [7] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. Zbl0475.28009

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