Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence
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| Format: | Preprint |
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2025
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| _version_ | 1866917032752054272 |
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| author | Robertson, Steven Aranov, Noy Soffer |
| author_facet | Robertson, Steven Aranov, Noy Soffer |
| contents | This is the first of a pair of papers, whose collective goal is to disprove a conjecture of Kemarsky, Paulin, and Shapira (KPS) on the escape of mass of Laurent series. This paper lays the foundations on which its sibling builds.
In particular, the $p$-Cantor sequence is introduced. This generalises the classical Cantor sequence into a $p$-automatic sequence for any odd prime $p$. Two main results are then established, both of which play a key role in the disproof of the KPS conjecture.
First, the two-dimensional sequence comprised of the Toeplitz determinants of the $p$-Cantor sequence over $\mathbb{F}_p$ is extensively studied. Indeed, the so-called profile of this sequence (which encodes the zero regions) is shown to be [p,p]-automatic. In the process of deriving this, the theory of so-called number walls is developed greatly. Many of these results are stated in full generality, as the authors expect them to be useful when tackling similar problems going forward.
Secondly, a natural process is described that converts number wall of an automatic sequence into a unique fractal. When this sequence is the aforementioned $p$-Cantor sequence, this fractal is shown to have Hausdorff dimension $\log((p^2+1)/2)/\log(p).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence Robertson, Steven Aranov, Noy Soffer Number Theory Dynamical Systems This is the first of a pair of papers, whose collective goal is to disprove a conjecture of Kemarsky, Paulin, and Shapira (KPS) on the escape of mass of Laurent series. This paper lays the foundations on which its sibling builds. In particular, the $p$-Cantor sequence is introduced. This generalises the classical Cantor sequence into a $p$-automatic sequence for any odd prime $p$. Two main results are then established, both of which play a key role in the disproof of the KPS conjecture. First, the two-dimensional sequence comprised of the Toeplitz determinants of the $p$-Cantor sequence over $\mathbb{F}_p$ is extensively studied. Indeed, the so-called profile of this sequence (which encodes the zero regions) is shown to be [p,p]-automatic. In the process of deriving this, the theory of so-called number walls is developed greatly. Many of these results are stated in full generality, as the authors expect them to be useful when tackling similar problems going forward. Secondly, a natural process is described that converts number wall of an automatic sequence into a unique fractal. When this sequence is the aforementioned $p$-Cantor sequence, this fractal is shown to have Hausdorff dimension $\log((p^2+1)/2)/\log(p).$ |
| title | Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2510.19449 |