# On normal numbers mod $2$

Colloquium Mathematicae (1998)

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

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topAhn, 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

top- [1] G. H. Choe, Spectral types of uniform distribution, Proc. Amer. Math. Soc. 120 (1994), 715-722. Zbl0803.11039
- [2] G. H. Choe, Ergodicity and irrational rotations, Proc. Roy. Irish Acad. 93A (1993), 193-202. Zbl0807.28009
- [3] R. B. Kirk, Sets which split families of measurable sets, Amer. Math. Monthly 79 (1972), 884-886. Zbl0249.28003
- [4] K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.
- [5] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1986.
- [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] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. Zbl0475.28009

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