Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence

TL;DR

The paper studies the p-Cantor sequence, introducing a two-dimensional framework via Toeplitz determinants and number walls to tackle the Kemarsky–Paulin–Shapira conjecture on escape of mass. It establishes that the right-side number wall of the p-Cantor sequence is 2D automatic and encodes this structure through a finite morphism on a 12-letter alphabet, yielding a precise profile under a [1,1]-coding. A fractal construction is then linked to the number wall, enabling a rigorous computation of the Hausdorff dimension of the resulting fractal as $\log\left((p^2+1)/2\right)/\log p$. The results provide both a concrete counterexample mechanism to KPS and new general tools for analyzing number walls and automatic-fractal connections beyond this problem. Overall, the work advances understanding of 2D automaticity in number-theoretic contexts and establishes a concrete fractal-geometry bridge via number walls and Cantor-type automatic sequences.

Abstract

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 $\log((p^2+1)/2)/\log(p).$

Figures (45)

• Figure 1: The number wall of $\widetilde{\textbf{S}}_r$. Each entry is either denoted with a number (where it is known) or a coloured dot (where it depends on $\textbf{S}$). The sequence $(s_i)_{0\le i <k^r}$ is coloured in light green. The other colours serve only to distinguish one part of the number wall from another.
• Figure 2: Illustration of a window in a number wall. The window, inner frame and outer frame are in red, green and blue, respectively.
• Figure 3: Each dot represents an entry in a number wall. The finite sequence (light green dots) that generates the finite number wall (whole picture) is on row zero. Each dot represents an entry in the finite number wall, with the dark green dots being those that are known explicitly and the black dots being those that are still variables.
• Figure 4: The finite number wall of a sequence over $\mathbb{F}_5$. The zero entries are in red, whereas the numbers 1-4 are illustrated in progressively darker shades of gray. The top row has index $-2$.
• Figure 5: Left: The number wall $W_5(\textbf{S})$ for the sequence $\textbf{S}=(1,1,3,2,1,0,0,0,2,0,2,0)\in\mathbb{F}_5^{12}$, where the zeroes are coloured in red. Right: The profile of $W_5(\textbf{S})$.

Theorems & Definitions (97)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Definition 1.5
• Example 1.6
• Conjecture 1.7: RG
• Theorem 1.8: Square Window Theorem, Lunnon, L09
• Theorem 1.9: Main Result 2
• Definition 2.1