Almost sure limit theorems with applications to non-regular continued fraction algorithms
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| Format: | Preprint |
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2023
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| _version_ | 1866911121138515968 |
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| author | Bonanno, Claudio Schindler, Tanja I. |
| author_facet | Bonanno, Claudio Schindler, Tanja I. |
| contents | We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},μ)$ with $μ$ a $σ$-finite measure and $μ(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from results by Aaronson that in general the asymptotic behaviour of the Birkhoff sums $S_Ng(x):= \sum_{j=1}^N\, (g\circ T^{j-1})(x)$ strongly depends on the point $x\in X$, and that there exists no sequence $(d_N)$ for which $S_Ng(x)/d_N \to 1$ for $μ$-almost every $x\in X$. In this paper we consider the case $g\not\in L^1(X,μ)$ assuming that there exists $E\in\mathcal{B}$ with $μ(E)<\infty$ and $\int_E g\,\mathrm{d}μ=\infty$ and continue the investigation initiated in previous work by the authors. We show that for transformations $T$ with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of $S_Ng(x)$ for an unbounded observable $g$ may be obtained using two methods, adding a number of summands depending on $x$ to $S_Ng$ and trimming. The obtained sums are then asymptotic to a scalar multiple of $N$. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or Rényi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_01132 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost sure limit theorems with applications to non-regular continued fraction algorithms Bonanno, Claudio Schindler, Tanja I. Dynamical Systems Number Theory 37A40, 37A25, 60F15, 11K50 We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},μ)$ with $μ$ a $σ$-finite measure and $μ(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from results by Aaronson that in general the asymptotic behaviour of the Birkhoff sums $S_Ng(x):= \sum_{j=1}^N\, (g\circ T^{j-1})(x)$ strongly depends on the point $x\in X$, and that there exists no sequence $(d_N)$ for which $S_Ng(x)/d_N \to 1$ for $μ$-almost every $x\in X$. In this paper we consider the case $g\not\in L^1(X,μ)$ assuming that there exists $E\in\mathcal{B}$ with $μ(E)<\infty$ and $\int_E g\,\mathrm{d}μ=\infty$ and continue the investigation initiated in previous work by the authors. We show that for transformations $T$ with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of $S_Ng(x)$ for an unbounded observable $g$ may be obtained using two methods, adding a number of summands depending on $x$ to $S_Ng$ and trimming. The obtained sums are then asymptotic to a scalar multiple of $N$. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or Rényi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms. |
| title | Almost sure limit theorems with applications to non-regular continued fraction algorithms |
| topic | Dynamical Systems Number Theory 37A40, 37A25, 60F15, 11K50 |
| url | https://arxiv.org/abs/2304.01132 |