FCFG-A: An Arithmetic Extension of Fractal Continued-Fraction Geometry

Author: Zhang Suhang (Luoyang, Henan)

Abstract

Within the framework of Fractal-Continued-Fraction Geometry (FCFG), this paper elevates finite continued fractions from “numerical objects” to “arithmetic recursive systems” and explores their structural correspondence with geometric recursive systems. We define arithmetic recursive systems, recursive operator chains, and structural mappings, propose the basic correspondence hypothesis of FCFG-A, and discuss the shared origin of arithmetic recursion and geometric recursion in the finite-order case. This paper further explores a unified interpretation of the Euclidean algorithm, Diophantine approximation, quadratic irrationals, Ramanujan continued fractions, and the continued-fraction expansions of the constants e and \pi within this operator-chain framework.

Keywords: fractal continued-fraction geometry; arithmetic recursion; continued fractions; recursive operators; structural correspondence

1. Introduction

The earlier framework of Fractal-Continued-Fraction Geometry (FCFG) proposed a correspondence between finite continued fractions and finite self-similar fractals:

r_n=S_n

where r_n is an n-th-order finite continued fraction and S_n is the global similarity ratio of an n-th-order finite self-similar fractal. This correspondence indicates that arithmetic recursion and geometric recursion share the same hierarchical structure in the finite-order case.

However, the existing FCFG mainly treats r_n as a numerical result and does not explicitly unfold its underlying operator-recursive structure. The goal of this paper is to elevate continued fractions from “numerical values” to “arithmetic recursive operator chains” and to establish a structural mapping between them and geometric scaling chains. This extension is called FCFG-A (FCFG-Arithmetic).

This paper does not claim to establish an independent new discipline; rather, it makes the arithmetic side explicit so that the basic correspondence of FCFG obtains a clearer algebraic expression.

2. Preliminaries and Review of the Core of FCFG

2.1 Finite Continued Fractions

Let a_1,a_2,\dots,a_n be a sequence of positive integers. Define the n-th-order finite continued fraction:

r_n=\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_n}}}

and denote r_n\in\mathbb{Q}.

2.2 Finite Self-Similar Fractals

Let F_0 be the initial figure. Recursively scale by the ratio 1/a_k and copy to obtain F_k. Define the global equivalent similarity ratio of F_n as S_n.

2.3 Basic Correspondence Hypothesis of FCFG

In the finite-order case, there exists a correspondence between the result of arithmetic recursion and the geometric equivalent similarity ratio: r_n=S_n. (Note: this serves as the starting point of the framework rather than a Euclidean axiom.)

3. Arithmetic Recursive Systems (Core Construction of FCFG-A)

Definition 3.1 (Arithmetic Recursive System)

Let a_1,\dots,a_n be a sequence of positive integers. Define the arithmetic recursive system \mathcal{A}_n=(a_1,\dots,a_n), corresponding to the continued-fraction value r_n.

Definition 3.2 (Recursive Operator)

For a positive integer a, define the nonlinear recursive operator:

T_a(x)=\frac{1}{a+x}

Then the arithmetic recursive system can be rigorously represented as an operator composition chain:

r_n=T_{a_1}\circ T_{a_2}\circ\cdots\circ T_{a_n}(0)

Definition 3.3 (Arithmetic Recursive Structure)

The ordered operator chain \mathcal{T}_n=(T_{a_1},T_{a_2},\dots,T_{a_n}) is called an n-th-order arithmetic recursive structure.

4. Structural Mapping Between Arithmetic and Geometry

Definition 4.1 (Structural Mapping)

For any n-th-order arithmetic recursive system \mathcal{A}_n, there exists an n-th-order geometric scaling chain \mathcal{R}_n=(R_{a_1},R_{a_2},\dots,R_{a_n}), where R_{a_k} denotes scaling by the ratio 1/a_k.

Define the mapping \phi:T_{a_k}\mapsto R_{a_k}. Since this mapping preserves the hierarchy and composition order of the recursion, that is,

\phi(T_{a_1}\circ\cdots\circ T_{a_n})

=

R_{a_1}\circ\cdots\circ R_{a_n},

we say that \phi establishes a structural correspondence between the arithmetic recursive chain and the geometric scaling chain. Combined with the hypothesis in 2.3, this correspondence satisfies r_n=S_n.

5. Discussion of Basic Properties (Based on Operator Chains)

Property 5.1 (Correspondence of Convergence)

By the definition r_n=S_n, the numerical convergence of the operator chain T_{a_k} is completely equivalent to the existence of the limit of the geometric scaling chain R_{a_k}.

Property 5.2 (Correspondence of Periodicity)

If the coefficient sequence \{a_k\} is eventually periodic, then, by Lagrange’s theorem, the arithmetic side r_n converges to a quadratic irrational; on the geometric side, since the scaling rule repeats across levels, the corresponding limit attractor F_\infty exhibits strict self-similarity and periodic topological features.

Property 5.3 (Dimension of Nonlinear IFS)

If \{a_k\} is bounded, then the operator chain T_{a_k} constitutes a compact hyperbolic iterated function system (IFS). It should be noted that, since T_a(x) is a fractional linear transformation (nonlinear), the Hausdorff dimension of its limit set cannot be obtained simply by applying the formula for linear IFS. Its dimension must be computed by numerical methods (such as the box-counting method) or by numerically solving the Moran equation.

6. FCFG-A Interpretation of Classical Arithmetic Structures

Based on the operator T_a(x), we can incorporate classical number-theoretic structures into the perspective of FCFG-A:

1. Euclidean algorithm: The recursive process of repeated division with remainder is essentially the process of generating the operator sequence \{T_{a_k}\}.

2. Diophantine approximation: The generation of the continued-fraction convergents p_k/q_k corresponds to evaluating the composition of the first k steps of the operator chain.

3. Ramanujan continued fractions: For example,

\frac{4}{\pi}=1+\frac{1^2}{2+\frac{3^2}{2+\dots}},

whose partial quotients exhibit regular growth. In the operator chain, this corresponds to a geometric compression process with “variable scale,” which is expected to correspond to special multifractal structures.

4. Constants e and \pi: Their partial-quotient sequences (2,1,2,1,1,4,\dots) and (3,7,15,1,292,\dots) correspond respectively to two specific infinite composite operator chains, determining their complexity under the geometric limit.

7. Discussion and Limitations

7.1 Relationship with FCFG

FCFG-A is an explicit formulation of the arithmetic side of FCFG. FCFG deals with the correspondence of numerical similarity ratios, while FCFG-A unfolds arithmetic operator chains, making them convenient for dynamical-system analysis.

7.2 Limitations

· This paper discusses only operator chains with positive coefficients, finite order, and convergence.

· A rigorous dimension calculation for nonlinear IFS has not yet been given in this paper; this is the core task of subsequent numerical experiments.

· Not all arithmetic recursions can be directly mapped to simple geometric scalings; complex variable-coefficient continued fractions correspond to variable-scale fractals.

8. Conclusion

Within the FCFG framework, by defining the operator T_a(x)=\frac{1}{a+x}, this paper establishes a strict structural correspondence between continued-fraction arithmetic recursion and geometric scaling. Through the operator-chain perspective, the Euclidean algorithm, quadratic irrationals, and Ramanujan continued fractions obtain a unified algebraic and geometric interpretation. This work lays a pure-mathematical foundation for subsequent numerical computation and dynamical-system analysis.

References

[1] Khintchine, A. Y. Continued Fractions. Dover Publications, 1964.

[2] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications. Wiley, 2014.

[3] Milnor, J. Dynamics in One Complex Variable. Princeton University Press, 2006.

[4] Berndt, B. C. Ramanujan’s Notebooks, Part III. Springer, 1991.