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ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 2 / October 2020
- Published online by Cambridge University Press:
- 13 February 2020, pp. 196-206
- Print publication:
- October 2020
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- Article
- Export citation
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For an irrational number
$x\in [0,1)$, let 
$x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients 
$\{a_{n}(x):n\geq 1\}$. Given 
$\unicode[STIX]{x1D6E9}\in \mathbb{N}$, for 
$n\geq 1$, the 
$n$th longest block function of 
$x$ with respect to 
$\unicode[STIX]{x1D6E9}$ is defined by 
$L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$, which represents the length of the longest consecutive sequence whose elements are all 
$\unicode[STIX]{x1D6E9}$ from the first 
$n$ partial quotients of 
$x$. We consider the growth rate of 
$L_{n}(x,\unicode[STIX]{x1D6E9})$ as 
$n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.
