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On the arithmetic of multidimensional continued fractions

Piotr Miska, Nadir Murru, Giuliano Romeo

TL;DR

This work addresses the problem of performing arithmetic on multidimensional continued fractions ($MCFs$). It generalizes Gosper's linear and bilinear transform framework to $MCFs$ by formulating Möbius and bilinear transforms in the Jacobi-Perron setting and providing concrete algorithms (GosperMCF1 and GosperMCF2) with feasibility proofs. The paper demonstrates, both theoretically and experimentally, that these transforms can be computed in a finite number of steps under appropriate independence assumptions, and reports linear growth in the number of inputs required relative to the number of outputs. It also proposes partial-output techniques to reduce intermediate growth and improve efficiency, highlighting practical potential for enabling sums and products of $MCFs$ in Diophantine approximation and Hermite-type problems.

Abstract

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the Möbius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms.

On the arithmetic of multidimensional continued fractions

TL;DR

This work addresses the problem of performing arithmetic on multidimensional continued fractions (). It generalizes Gosper's linear and bilinear transform framework to by formulating Möbius and bilinear transforms in the Jacobi-Perron setting and providing concrete algorithms (GosperMCF1 and GosperMCF2) with feasibility proofs. The paper demonstrates, both theoretically and experimentally, that these transforms can be computed in a finite number of steps under appropriate independence assumptions, and reports linear growth in the number of inputs required relative to the number of outputs. It also proposes partial-output techniques to reduce intermediate growth and improve efficiency, highlighting practical potential for enabling sums and products of in Diophantine approximation and Hermite-type problems.

Abstract

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the Möbius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms.
Paper Structure (8 sections, 112 equations, 8 figures, 4 algorithms)

This paper contains 8 sections, 112 equations, 8 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Classical Gosper's algorithm revised
  3. Möbius transform of multidimensional continued fractions.
  4. Proof of feasibility of the algorithm
  5. Sums and products of MCFs
  6. Proof of feasibility of the algorithm
  7. Experimental results
  8. Possible modification of the algorithm - partial output

Figures (8)

  • Figure 1: Mean number of inputs required by Algorithm \ref{['Alg: GospMCF1']} for $m=2$ to obtain outputs up to $10$, for $(\sqrt[3]{d},\sqrt[3]{d^2})$, $2\leq d\leq 100$ non-cube integer, using $C_1$, $C_2$, $C_3$ as defined in \ref{['Eq: CCC']}.
  • Figure 2: Number of inputs required by Algorithm \ref{['Alg: GospMCF1']} for $m=2$ to obtain outputs up to $10$, for $(\sqrt[3]{d},\sqrt[3]{d^2})$, $d=2,3,5$, using $C_1$ as defined in \ref{['Eq: CCC']}
  • Figure 3: Mean number of inputs required by Algorithm \ref{['Alg: GospMCF1']} for $m=2$ up to $1000$ outputs, for $4$ different $3\times 3$ matrices with entries in $\{0,\ldots,1000\}$. The mean is computed among $1000$ different random input sequences $((a_n),(b_n))$ with $b_n\leq a_n\leq 1000$.
  • Figure 4: Number of inputs required by Algorithm \ref{['Alg: GospMCF1']} for $m=2$ to get up to $10000$ outputs, for $1000$ different random sequences of partial quotients in $\{0,\ldots,10000\}$ and for $10000$ random transformations with coefficients in $\{0,\ldots,10000\}$.
  • Figure 5: Number of inputs required by Algorithm \ref{['Alg: GospMCF1']} for $m=2$ to get up to $2000$ outputs for $500$ different random sequences. The partial quotients are randomly selected in $\{0,\ldots,10\}$ for the left graphic and in $\{0,\ldots,1000000\}$ for the right graphic.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Example 2.1
  • Example 6.1
  • Example 6.2
  • Example 6.3