# Walking on Real Numbers

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## Keywords

Random Walk Rational Number Continue Fraction Mathematical Intelligencer Simple Random Walk
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## References

- [1]S. Albeverioa, M. Pratsiovytyie, and G. Torbine G, “Topological and fractal properties of real numbers which are not normal”.
*Bulletin des Sciences Mathématiques*,**129**(2005), 615–630.Google Scholar - [2]J.-P. Allouche and J. Shallit,
*Automatic Sequences: Theory, Applications, Generalizations*. Cambridge University Press, Cambridge, 2003.Google Scholar - [3]D. H. Bailey and J. M. Borwein, “Normal numbers and pseudorandom generators,”
*Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday*, Springer, 2012, in press.Google Scholar - [4]D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, “An empirical approach to the normality of pi”.
*Experimental Mathematics*, 2012; in press.Google Scholar - [5]D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance. “On the binary expansions of algebraic numbers”.
*Journal of Number Theory Bordeaux*,**16**(2004), 487–518.Google Scholar - [6]M. Barnsley,
*Fractals Everywhere*, Academic Press, Inc., Boston, MA, 1988.Google Scholar - [7]D. H. Bailey, P. B. Borwein, and S. Plouffe, “On the rapid computation of various polylogarithmic constants”.
*Mathematics of Computation*,**66**, no. 218 (1997), 903–913.Google Scholar - [8]D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques and applications”.
*Mathematics of Computation*,**70**, no. 236 (2000), 1719–1736.Google Scholar - [9]D. H. Bailey and R. E. Crandall, “On the random character of fundamental constant expansions”.
*Experimental Mathematics*,**10**, no. 2 (2001), 175–190.Google Scholar - [10]D. H. Bailey and R. E. Crandall, “Random generators and normal numbers,”
*Experimental Mathematics*,**11**(2002), no. 4, 527–546.Google Scholar - [11]D. H. Bailey and M. Misiurewicz, “A strong hot spot theorem,”
*Proceedings of the American Mathematical Society*,**134**(2006), no. 9, 2495–2501.Google Scholar - [12]G. Barat, R. F. Tichy, and R. Tijdeman, Digital blocks in linear numeration systems.
*Number theory in progress*,**2**(Zakopane-Kościelisko, 1997), de Gruyter, Berlin (1999), 607–631.Google Scholar - [13]M. N. Barber and B. W. Ninham,
*Random and Restricted Walks: Theory and Applications*, Gordon and Breach, New York, 1970.Google Scholar - [14]A. Belshaw and P. B. Borwein, “Champernowne’s number, strong normality, and the X chromosome,”
*Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday*, Springer, 2012, in press.Google Scholar - [15]L. Berggren, J. M. Borwein, and P. B. Borwein,
*Pi: a Source Book*, Springer-Verlag, Third Edition, 2004.Google Scholar - [16]J. M. Borwein and D. H. Bailey,
*Mathematics by Experiment: Plausible Reasoning in the 21st Century*, 2nd ed., A. K. Peters, Natick, MA, 2008.Google Scholar - [17]J. Borwein, D. Bailey, N. Calkin, R. Girgensohn, R. Luke, V. Moll,
*Experimental Mathematics in Action*. A. K. Peters, Natick, MA, 2007.Google Scholar - [18]J. M. Borwein and P. B. Borwein,
*Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity*, John Wiley, New York, 1987, paperback 1998.Google Scholar - [19]J. M. Borwein, P. B. Borwein, R. M. Corless, L. Jörgenson, and N. Sinclair, “What is organic mathematics?”
*Organic mathematics*(Burnaby, BC, 1995), CMS Conf. Proc.,**20**, Amer. Math. Soc., Providence, RI, 1997, 1–18.Google Scholar - [20]P. B. Borwein, “On the irrationality of certain series.”
*Math. Proc. Cambridge Philos. Soc.***112**(1992) 141–146.Google Scholar - [21]P. B. Borwein and L. Jörgenson, “ Visible structures in number theory,”
*Amer. Math. Monthly***108**(2001), no. 10, 897–910.Google Scholar - [22]C. S. Calude, “Borel normality and algorithmic randomness,” in G. Rozenberg, A. Salomaa (eds.),
*Developments in Language Theory*, World Scientific, Singapore, 1994, 113–129.Google Scholar - [23]C.S. Calude,
*Information and Randomness: An Algorithmic Perspective*, 2nd ed., Revised and Extended, Springer-Verlag, Berlin, 2002.Google Scholar - [24]D. G. Champernowne, “The construction of decimals normal in the scale of ten.”
*Journal of the London Mathematical Society*,**8**(1933) 254–260.Google Scholar - [25]M. Coons, “(Non)automaticity of number theoretic functions,”
*J. Théor. Nombres Bordeaux*,**22**(2010), no. (2), 339–352.Google Scholar - [26]A. H. Copeland and P. Erdős, “Note on normal numbers,”
*Bulletin of the American Mathematical Society*,**52**(1946), 857–860.Google Scholar - [27]R. E. Crandall, “The googol-th bit of the Erdős–Borwein constant,”
*Integers*, A23, 2012.Google Scholar - [28]M. Dekking, M. Mendès France, and A. van der Poorten, “Folds,”
*Math. Intelligencer***4**(1982), no. 3, 130–138.Google Scholar - [29]M. Dekking, M. Mendès France, and A. van der Poorten, “Folds II,”
*Math. Intelligencer***4**(1982), no. 4, 173–181.Google Scholar - [30]M. Dekking, M. Mendès France, and A. van der Poorten, “Folds III,”
*Math. Intelligencer***4**(1982), no. (4), 190–195.Google Scholar - [31]D. Y. Downham and S. B. Fotopoulos, “The transient behaviour of the simple random walk in the plane,”
*J. Appl. Probab.***25**(1988), no. 1, 58–69.Google Scholar - [32]D. Y. Downham and S. B. Fotopoulos, “A note on the simple random walk in the plane,”
*Statist. Probab. Lett.*, 17 (1993), no. 3, 221–224.Google Scholar - [33]A. Dvoretzky and P. Erdős, “Some problems on random walk in space,”
*Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability*, (1951), 353–367.Google Scholar - [34]P. Hertling, “Simply normal numbers to different bases,”
*Journal of Universal Computer Science*,**8**, no. 2 (2002), 235–242.Google Scholar - [35]H. J. Jeffrey, Chaos game representation of gene structure,
*Nucl. Acids Res.***18**no 2, (1990) 2163–2170.Google Scholar - [36]B. D. Hughes,
*Random Walks and Random Environments*,*Vol. 1. Random Walks*, Oxford Science Publications, New York, (1995).Google Scholar - [37]H. Kaneko, “On normal numbers and powers of algebraic numbers,”
*Integers*,**10**(2010), 31–64.Google Scholar - [38]D. Khoshnevisan, “Normal numbers are normal,”
*Clay Mathematics Institute Annual Report*(2006), 15 & 27–31.Google Scholar - [39]G. Marsaglia, “On the randomness of pi and other decimal expansions,” preprint, 2010.Google Scholar
- [40]G. Martin, “Absolutely abnormal numbers,”
*Amer. Math. Monthly*,**108**(2001), no. 8, 746-754.Google Scholar - [41]J. Mah and J. Holdener, “When Thue–Morse meets Koch,”
*Fractals*,**13**(2005), no. 3, 191–206.Google Scholar - [42]S. M. Ross,
*Stochastic Processes*. John Wiley & Sons, New York, 1983.Google Scholar - [43]R. Stoneham, “On absolute \((j, \varepsilon)\)-normality in the rational fractions with applications to normal numbers,”
*Acta Arithmetica*,**22**(1973), 277–286.Google Scholar - [44]M. Queffelec, “Old and new results on normality,”
*Lecture Notes – Monograph Series*,**48**,*Dynamics and Stochastics*, 2006, Institute of Mathematical Statistics, 225–236.Google Scholar - [45]
- [46]A. J. Yee, “y-cruncher-multi-threaded pi program,” http://www.numberworld.org/y-cruncher, 2010.
- [47]A. J. Yee and S. Kondo, “10 trillion digits of pi: A case study of summing hypergeometric series to high precision on multicore systems,” preprint, 2011, available at http://hdl.handle.net/2142/28348.

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