Schneider Continued Fraction Expansion

• Schneider continued fraction expansion is a p-adic analogue of classical continued fractions, using homographic transformations with digit-exponent pairs.
• It constructs convergents via a recursive matrix recurrence that provides explicit complexity bounds akin to the classical Lame theorem.
• The associated dynamical system exhibits ergodicity, exponential convergence, and multifractal properties useful in spectral and geometric analyses of p-adic sets.

The Schneider continued fraction expansion is a formal p-adic analogue of classical continued fractions in the field of $p$-adic numbers $\mathbb{Q}_p$. It is constructed via a specific homographic transformation using pairs of digits and exponents, resulting in convergents that rapidly approximate p-adic numbers. The expansion induces a dynamical system with intrinsic ergodic and multifractal properties and provides foundational bounds on the complexity of rational expansions, paralleling the classical Lame theorem for real continued fractions. Schneider expansions are instrumental in the spectral analysis and geometric dimension theory of p-adic sets.

1. Notation, Preliminaries, and p-adic Framework

Let $p \ge 3$ be a fixed prime. The field $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value $|\cdot|_p$, defined by $|x|_p = p^{-v_p(x)}$, where $v_p(x)$ is the $p$-adic valuation. Each $x \in \mathbb{Q}_p$ admits a unique digit expansion

$$x = \sum_{i=-\infty}^{\infty} a_i p^i,\quad a_i \in \{0,1,\dots,p-1\}.$$

The Schneider algorithm exploits this ultrametric representation to encode continued fractions efficiently for both rationals and general $p$-adic numbers (Belhadef et al., 2024).

2. Formal Definition of Schneider Expansion

For $x \in \mathbb{Q}_p$, the Schneider continued fraction is constructed from a sequence of pairs $(a_0,\alpha_0), (a_1,\alpha_1), \dots$ with $a_i \in \{1,\dots,p-1\}$ and $\alpha_i \in \mathbb{N}$. The expansion is written formally as

$$[\, (a_0,\alpha_0);\, (a_1,\alpha_1), \dots, (a_n, \alpha_n); x \,],$$

and defined recursively by the homographic rule: $[\, (a,\alpha); x \,] = a + \frac{p^{\alpha}}{x},$ with the iterative relation

$$[\, (a_0,\alpha_0);(a_1,\alpha_1), \dots, (a_n, \alpha_n); x\,] = [\, (a_0,\alpha_0);(a_1,\alpha_1), \dots, (a_{n-1}, \alpha_{n-1}); [\, (a_n,\alpha_n); x \,]\,].$$

The $n$th convergent is explicitly rational, given as $U_n/W_n$, where $(U_n, W_n)$ are determined by matrix products: $$M(a,\alpha) = \begin{pmatrix} a p^{\alpha} & 1 \ 1 & 0 \end{pmatrix},\qquad M(a_0,\alpha_0)\cdots M(a_n,\alpha_n).$$

3. Schneider Algorithm: Construction for Rationals

Given $r = a/b \in \mathbb{Q}$ with $(a, b, p) = 1$, the algorithm initializes with $y_{-1} = a$, $y_0 = b$, and proceeds inductively:

• For $m \ge 0$, select $b_m \in \{0,1,\dots,p-1\}$ and $\alpha_m \in \mathbb{N}$ so that

$y_{m-1} - b_m\, y_m = p^{\alpha_m} y_{m+1},$

with $y_{m+1}$ coprime to $p$ and $y_m$. The pair $(b_m, \alpha_m)$ constitutes the partial quotient.

• The process either terminates (if $y_{N+1}=0$) or becomes stationary with $(b_m,\alpha_m) = (p-1,1)$ for all large $m$ by Bundschuh’s theorem (Belhadef et al., 2024).

4. Complexity, Matrix Recurrence, and Length Bounds

The length $k+1$ of the non-stationary part in the Schneider expansion for $r$ is given by the main theorem (Belhadef et al., 2024): $$k = \left\lceil \frac{\ln|\theta|}{\ln T_1} \right\rceil + 1,$$ where

$$\theta = \frac{(T_1 - p^{\alpha})(a - b T_1)}{(T_2 - p^{\alpha})(a - b T_2)},$$

and $T_1, T_2$ are the real roots of $T^2 - a_* T - p^{\alpha} = 0$ for the limiting partial quotient $a_* = p-1$. The matrix recurrence for convergents is

$$U_n = a\, U_{n-1} + p^{\alpha} U_{n-2},\quad U_{-1}=1,\ U_0=a.$$

This provides explicit logarithmic bounds, establishing a $p$-adic analogue to the real Lame theorem for continued fractions.

5. Ergodic and Convergence Properties

The Schneider map $T_{Sch}$ acts on $p\mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p < 1\}$ by

$T_{Sch}(x) = \frac{p^{v_p(x)}}{x} - b_1(x),$

where $b_1(x) \in \{1,\dots,p-1\}$ with $p^{v_p(x)}/x \equiv b_1(x) \pmod p$. The induced dynamical system preserves the Haar measure and exhibits ergodicity and mixing (Rao et al., 2020). The transformation expands cylinder sets by $p^{a_1(x)}$ and the recurrence yields exponential convergence of convergents $P_n/Q_n$: $|x - P_n/Q_n|_p \le p^{-n},$ indicating rapid $p$-adic approximation.

6. Lyapunov Spectrum and Multifractal Analysis

The thermodynamic formalism applied to the Schneider map yields a multifractal (Lyapunov) spectrum describing the dimension of level sets defined by the limit

$$\lambda(x) = \log p \cdot \lim_{n \to \infty} \frac{a_1(x) + \cdots + a_n(x)}{n}$$

for admissible $x$. The precise Hausdorff dimension $L(\alpha)$ of the set of $x$ with $\lambda(x)=\alpha$ is (Alvarado et al., 9 Jan 2026): $$L(\alpha) = \frac{1}{\alpha} \left[ \log(p-1) + \log(\alpha - \log p) - \log \log p + \alpha \log_p \alpha - \alpha \log_p (\alpha - \log p) \right],\quad \alpha \ge \log p.$$ This analytic formula characterizes the multifractal geometry of $p$-adic numbers by their expansion rates and relates directly to rational approximation speed via Schneider convergents.

7. Examples and Theoretical Analogies

Explicit examples illustrate the non-stationary part of the Schneider expansion:

• For $p=3$, $r = 2/5$: $(1,1), (1,1), (1,1), (2,1), (2,1),\dots$, with $k=3$ as predicted.
• For $p=3$, $r = 1259/701$: $(1,2), (1,2), (1,2), (1,2), (1,2)$ before stabilization to $(2,1)$ (Belhadef et al., 2024).

The analogy with Browkin continued fraction expansions is direct: both admit $p$-adic Lame-type length bounds and matrix recurrence complexity, generalizing the real case ($O(\ln b)$) to $p$-adic rational expansions. The multifractal formalism for Schneider expansions parallels the thermodynamic theory for the real Gauss map, with the ultrametric context permitting analytic closed-form spectra unparalleled in the real setting (Alvarado et al., 9 Jan 2026).