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ZAREMBA, SALEM AND THE FRACTAL NATURE OF GHOST DISTRIBUTIONS

Part of: Classical measure theory Sequences and sets

Published online by Cambridge University Press:  06 October 2022

MICHAEL COONS ,
JAMES EVANS ,
ZACHARY GROTH  and
NEIL MAÑIBO
MICHAEL COONS*
Affiliation:
Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA
JAMES EVANS
Affiliation:
School of Information and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia e-mail: james.evans10@uon.edu.au
ZACHARY GROTH
Affiliation:
School of Information and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia e-mail: zachary.groth@uon.edu.au
NEIL MAÑIBO
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany e-mail: cmanibo@math.uni-bielefeld.de
*
e-mail: mjcoons@csuchico.edu
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Abstract

Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem’s strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.

Keywords

automatic sequenceregular sequenceaperiodic ordersymbolic dynamicsdilation equationself-affine setcontinuous measure

MSC classification

Primary: 11B85: Automata sequences
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 374 - 389
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Allouche, J.-P. and Shallit, J., ‘The ring of $k$ -regular sequences’, Theoret. Comput. Sci. 98(2) (1992), 163–197.CrossRefGoogle Scholar
Baake, M. and Coons, M., ‘A natural probability measure derived from Stern’s diatomic sequence’, Acta Arith. 183(1) (2018), 87–99.CrossRefGoogle Scholar
Benedetto, J. J., Bernstein, E. and Konstantinidis, I., ‘Multiscale Riesz products and their support properties’, Acta Appl. Math. 88(2) (2005), 201–227.CrossRefGoogle Scholar
Billingsley, P., Probability and Measure, 3rd edn (John Wiley & Sons, New York, 1995).Google Scholar
Bourgain, J. and Kontorovich, A., ‘On Zaremba’s conjecture’, Ann. of Math. (2) 180(1) (2014), 137–196.CrossRefGoogle Scholar
Coons, M., ‘Mahler takes a regular view of Zaremba’, Integers 18A (2018), Article no. A6, 15 pages.Google Scholar
Coons, M., Evans, J. and Mañibo, N., ‘Spectral theory of regular sequences’, Doc. Math. 27 (2022), 629–653.CrossRefGoogle Scholar
Coons, M. and Spiegelhofer, L., ‘Number theoretic aspects of regular sequences’, in: Sequences, Groups, and Number Theory (eds. V. Berthé and M. Rigo), Trends in Mathematics (Birkhäuser/Springer, Cham, 2018), 37–87.CrossRefGoogle Scholar
Daubechies, I. and Lagarias, J. C., ‘Two-scale difference equations. I. Existence and global regularity of solutions’, SIAM J. Math. Anal. 22(5) (1991), 1388–1410.CrossRefGoogle Scholar
Daubechies, I. and Lagarias, J. C., ‘Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals’, SIAM J. Math. Anal. 23(4) (1992), 1031–1079.CrossRefGoogle Scholar
Dumas, P., ‘Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences’, Linear Algebra Appl. 438(5) (2013), 2107–2126.CrossRefGoogle Scholar
Dumas, P., ‘Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated’, Theoret. Comput. Sci. 548 (2014), 25–53.CrossRefGoogle Scholar
Eilenberg, S., Automata, Languages, and Machines, Vol. A (Academic Press, New York, 1974).Google Scholar
Falconer, K., Fractal Geometry, 3rd edn (John Wiley & Sons, Chichester, 2014).Google Scholar
Furstenberg, H. and Kesten, H., ‘Products of random matrices’, Ann. Math. Stat. 31 (1960), 457–469.CrossRefGoogle Scholar
Huang, S. Y., ‘An improvement to Zaremba’s conjecture’, Geom. Funct. Anal. 25(3) (2015), 860–914.CrossRefGoogle Scholar
Jenkinson, O. and Pollicott, M., ‘Rigorous dimension estimates for Cantor sets arising in Zaremba theory’, in: Dynamics: Topology and Numbers (eds. P. Moree, A. Pohl, L. Snoha and T. Ward), Contemporary Mathematics, 744 (American Mathematical Society, Providence, RI, 2020), 83–107.CrossRefGoogle Scholar
Jungers, R., The Joint Spectral Radius, Lecture Notes in Control and Information Sciences, 385 (Springer-Verlag, Berlin, 2009).CrossRefGoogle Scholar
Rota, G.-C. and Strang, G., ‘A note on the joint spectral radius’, Indag. Math. (N.S.) 22 (1960), 379–381.CrossRefGoogle Scholar
Salem, R., ‘On some singular monotonic functions which are strictly increasing’, Trans. Amer. Math. Soc. 53 (1943), 427–439.CrossRefGoogle Scholar
Varjú, P. P., ‘On the dimension of Bernoulli convolutions for all transcendental parameters’, Ann. of Math. (2) 189(3) (2019), 1001–1011.CrossRefGoogle Scholar
Zaremba, S. K., ‘La méthode des “bons treillis” pour le calcul des intégrales multiples’, in: Applications of Number Theory to Numerical Analysis (ed. S. K. Zaremba) (Academic Press, New York, 1972), 39–119.CrossRefGoogle Scholar