A normal number theorem for Brownian motion
Mathematics Research Reports, Volume 2 (2021) , pp. 15-20.

We prove that we can generate “a lot” of random normal numbers (in the sense of full Hausdorff dimension), which are outside the scope of Borel’s Normal Number Theorem (in the sense of zero Lebesgue measure). The ingredient is to run Brownian motion over a specific Cantor-like time-set. These are closely related to an equidistributed property of some dynamical orbit with random inputs, which itself is new and significant.

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DOI: https://doi.org/10.5802/mrr.7
Classification: 37A05,  11K16,  11K36,  60G15,  60G17
Keywords: Brownian motion, fractional Brownian motion, equidistribution sequence, normal number, dynamical orbit
@article{MRR_2021__2__15_0,
     author = {Narn-Rueih Shieh},
     title = {A normal number theorem for {Brownian} motion},
     journal = {Mathematics Research Reports},
     pages = {15--20},
     publisher = {MathOA foundation},
     volume = {2},
     year = {2021},
     doi = {10.5802/mrr.7},
     language = {en},
     url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.7/}
}
Narn-Rueih Shieh. A normal number theorem for Brownian motion. Mathematics Research Reports, Volume 2 (2021) , pp. 15-20. doi : 10.5802/mrr.7. https://mrr.centre-mersenne.org/articles/10.5802/mrr.7/

[1] R. Balka; Y. Peres Uniform dimension results for fractional Brownian motion, J. Fractal Geom., Volume 4 (2017), pp. 147-183 | Article | MR 3667705 | Zbl 1376.60060

[2] H. Davenport; P. Erdős; W. LeVeque On Weyl’s criterion for uniform distribution, Michigan Math. J., Volume 10 (1963), pp. 311-314 | Article | MR 153656 | Zbl 0119.28201

[3] K. Falconer Fractal Geometry, Mathematical Foundations and Applications, 1990 | Article | Zbl 0689.28003

[4] J. Fraser; T. Sahlsten On the Fourier analytic structure of the Brownian graph., Analysis & PDE, Volume 11 (2018), pp. 115-132 | Article | MR 3707292 | Zbl 1388.60114

[5] G. Harman On hundred years of normal numbers, Surveys in Number Theory, Papers from The Millennial Conference on Number Theory (M.A. Bennett, ed.), CRC Press, 2003

[6] M. Hochman; P. Sherkin Equidistribution from fractal measures, Invent. Math., Volume 202 (2015), pp. 427-479 | Article | MR 3402802

[7] J. P. Kahane Some Random Series of Functions, Second Edition, Cambridge University Press, 1985

[8] D. Khoshnevisan Normal numbers are normal, The Clay Institute Report, 2006

[9] I. Laba Harmonic analysis and the geometry of fractals, Proceedings of the 2014 International Congress of Mathematicians, Seoul, 2014 | MR 3729030 | Zbl 1373.42006

[10] P. Mörters; Y. Peres Brownian Motion, Cambridge Univ. Press, 2010

[11] N.-R. Shieh; Y. Xiao Images of Gaussian random fields: Salem sets and interior points, Studia Math., Volume 176 (2006), pp. 37-60 | Article | MR 2263961 | Zbl 1105.60023

[12] S. J. Taylor On the connexion between Hausdorff measures and generalized capacity, Proc. Cambridge Philos. Soc., Volume 57 (1961), pp. 524-531 | Article | MR 133420 | Zbl 0106.26802