On points contain arithmetic progressions in their lüroth expansion

Zhang, Zhenliang; Cao, Chun-Yun

doi:10.1016/s0252-9602(15)30093-x

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Cited by 3 publications

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“…However, there do exist cases such that the inequality (1.1) is an equality, for example continued fractions [9], Lüroth expansions [10] and general d-decaying Gauss-like systems, by combining Theorem 1.2 above.…”

Section: Introductionmentioning

confidence: 99%

On Points With Positive Density of the Digit Sequence in Infinite Iterated Function Systems

Zhang

Cao

2016

J. Aust. Math. Soc.

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Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.

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Section: Introductionmentioning

confidence: 99%