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- W3099774138 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 $unicode[STIX]{x1D6FD}in (1,1.787ldots )$ then every $xin (0,1/(unicode[STIX]{x1D6FD}-1))$ has a simply normal $unicode[STIX]{x1D6FD}$ -expansion. We also prove that if $unicode[STIX]{x1D6FD}in (1,(1+sqrt{5})/2)$ then every $xin (0,1/(unicode[STIX]{x1D6FD}-1))$ 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." @default.
- W3099774138 created "2020-11-23" @default.
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- W3099774138 date "2018-12-19" @default.
- W3099774138 modified "2023-09-24" @default.
- W3099774138 title "Digit frequencies and self-affine sets with non-empty interior" @default.
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- W3099774138 doi "https://doi.org/10.1017/etds.2018.127" @default.
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