p-Cantor Sequence: Automaticity & Fractal Geometry

• p-Cantor sequence is a p-automatic infinite word generated by a uniform p-morphism, exhibiting self-similarity and fractal structure.
• Its associated Toeplitz determinants form a bi-dimensional number wall with [p,p]-automatic patterning and square window clustering of zeros.
• The sequence provides counterexamples to the KPS conjecture through partial escape of mass, offering valuable insights into arithmetic geometry over finite fields.

A $p$-Cantor sequence is a generalization of the classical Cantor sequence constructed over a prime field $\mathbb{F}_p$, with deep connections to automatic sequences, self-similar fractal geometry, Toeplitz determinants, and Diophantine phenomena in function fields. For each odd prime $p$, the $p$-Cantor sequence is defined as the unique fixed point of a uniform $p$-morphism acting on the digit alphabet $\{0,1,\ldots,p-1\}$, which produces a $p$-automatic, self-similar infinite word. The structure of its associated Toeplitz determinants in $\mathbb{F}_p$—arranged as a two-dimensional "number wall"—exhibits fractal properties and [p,p]-automaticity, allowing for rigorous analysis of arithmetic and geometric phenomena, such as escape of mass in Laurent series, in characteristic $p$.

1. Definition and Generation of the $p$-Cantor Sequence

The $p$-Cantor sequence, denoted $C^{(p)}$, is specified by a substitution (uniform $p$-morphism) $\varphi_p$: $$\varphi_p(n)_i = n \cdot \binom{p_2}{i/2} \mod p \quad \text{for even } i, \quad \varphi_p(n)_i = 0 \text{ otherwise}$$ where $p_2 = (p-1)/2$ and $n \in \{0,1,\ldots,p-1\}$.

Its fixed point is constructed as: $C^{(p)} = \lim_{k \to \infty}\varphi_p^k(1)$ This sequence is $p$-automatic and reflects the self-similarity property: at every scale, one can decompose the word of length $p^k$ into $p$ blocks, each generated via combinatorial coefficients. A symmetry property holds: $$c^{(p)}_i = c^{(p)}_{p^k-1-i} \quad \text{for all } i, k$$ emphasizing its palindromic and fractal features.

2. Toeplitz Determinants and the Number Wall

For any doubly-infinite sequence $S$, including $C^{(p)}$, the $p$-number wall is the bi-infinite array $W_p(S)$ given by Toeplitz determinants: $$W_p(S)[m,n] = \begin{cases} \det(T_S(n; m)), & m \geq 0 \ 1, & m = -1 \ 0, & m < -1 \ \end{cases}$$ where $T_S(n; m)$ is the $(m+1)\times(m+1)$ Toeplitz matrix with entries from $S$. A fundamental result, the Square Window Theorem, shows that zeros in $W_p(S)$ cluster in square blocks ("windows") surrounded by frames of nonzero elements.

The profile $\chi(W_p(C^{(p)}))$—which marks zeros and nonzeros—is shown to be $[p,p]$-automatic: it is generated by a two-dimensional uniform substitution on an alphabet of $12$ letters, followed by a coding that maps nonzero entries to a fixed symbol (e.g., $X$).

3. Fractal Structure from Toeplitz Determinants

By interpreting the number wall as a geometric object, the profile induces a self-similar fractal in $[0,1]^2$ via a covering construction: for each $r$, subdivide the square into $p^r \times p^r$ grids, including those corresponding to nonzero entries. The limiting set

$$\mathcal{A}_f(C^{(p)},\mathbb{F}_p) = \bigcap_{r=1}^{\infty}\mathcal{J}_r$$

is shown to have a precise Hausdorff dimension: $$\dim_H(\mathcal{A}_f(C^{(p)},\mathbb{F}_p)) = \frac{\log((p^2+1)/2)}{\log(p)}$$ This demonstrates a direct link between the combinatorial construction of $C^{(p)}$, the algebraic properties of Toeplitz determinants, and the emergent fractal geometry.

4. Escape of Mass and Counterexamples to the KPS Conjecture

Kemarsky, Paulin, and Shapira conjectured that any Laurent series over $\mathbb{F}_p$ would exhibit full escape of mass with respect to irreducible polynomials $P(t)$, meaning the limiting measure associated to partial quotients vanishes almost everywhere along certain dynamical progressions. The $p$-Cantor sequence, via its Laurent series $\Theta_p(t)=t^{-1}\sum_i c^{(p)}_i t^{-i}$, provides an explicit counterexample for odd $p$ (Aranov et al., 22 Oct 2025): the escape of mass is exactly $2/p$ (on appropriate subsequences of diagonals), not $1$.

The proof leverages the structure of the number wall—zeros correspond to large partial quotients in continued fraction expansions—showing that only partial escape of mass occurs (not full), but maximal and generic escape variants do hold under appropriate versions.

5. Automaticity and Self-Similarity

The $p$-Cantor sequence is $p$-automatic, meaning its elements are computed by a finite automaton reading the $p$-ary expansions of indices. This automaticity extends to its Hankel and Toeplitz determinants, facilitating deep arithmetic and analytic analysis. The self-similarity, both in its construction and its induced fractals, reflects the recursive combinatorial operations underlying the sequence.

6. Mathematical Significance and Broader Applications

These developments unify concepts from recursion theory, finite automata, combinatorics on words, harmonic analysis, and fractal geometry. The explicit description of the number wall tiling, automaticity in two dimensions, and computation of fractal dimension provide new tools for analyzing Diophantine approximation in positive characteristic, the distribution of continued fraction expansions, and problems in symbolic dynamics.

The $p$-Cantor sequence's properties—automaticity, fractal emergence, and arithmetic rigidity—make it a canonical test object for further investigations into the interaction between automata-theoretic constructions and arithmetic geometry over finite fields, as well as for exploring the structure of counterexamples to conjectures in escape of mass phenomena (Robertson et al., 22 Oct 2025).