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Digit frequencies and self-affine sets with non-empty interior

Part of: Elementary number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  19 December 2018

SIMON BAKER
SIMON BAKER*
Affiliation:
Mathematics institute, University of Warwick, Coventry, CV4 7AL, UK email [email protected]
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Abstract

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\unicode[STIX]{x1D6FD}\in (1,1.787\ldots )$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a simply normal $\unicode[STIX]{x1D6FD}$-expansion. We also prove that if $\unicode[STIX]{x1D6FD}\in (1,(1+\sqrt{5})/2)$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a $\unicode[STIX]{x1D6FD}$-expansion for which the digit frequency does not exist, and a $\unicode[STIX]{x1D6FD}$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1$, then every non-trivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and gives rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.

Keywords

expansions in non-integer basesdigit frequenciesself-affine sets

MSC classification

Primary: 11A63: Radix representation; digital problems
Secondary: 28A80: Fractals 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1755 - 1787
Copyright
© Cambridge University Press, 2018

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