General Recurrence Multidimensional Zeckendorf Representations

TL;DR

The paper extends Zeckendorf-type representations to a broad family of linear recurrences in a multidimensional setting by introducing $\vec{c}$-SRs for weakly decreasing recurrence vectors with $c_k=1$ and constructing the vector sequence $\vec{X}_n$ governed by $\vec{X}_n := c_1\vec{X}_{n-1} + \cdots + c_k\vec{X}_{n-k}$. It develops a carrying/borrowing framework to transform arbitrary $\vec{c}$-decompositions into $\vec{c}$-SRs, and proves existence and uniqueness via a bijection with scalar Positive Linear Recurrence Sequences (PLRS) through the map $S_n$. The multidimensional decompositions inherit key scalar properties, including Gaussian convergence for the number of summands and summand minimality when $\vec{c}$ is weakly decreasing. The work generalizes previous results (ABJ11, MW12) and opens avenues for further multidimensional $f$-decompositions and geometric/combinatorial analyses of region growth in $\mathbb{Z}^{k-1}$.

Abstract

We present a multidimensional generalization of Zeckendorf's Theorem (any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers) to a large family of linear recurrences. This extends work of Anderson and Bicknell-Johnson in the multi-dimensional case when the underlying recurrence is the same as the Fibonacci one. Our extension applies to linear recurrence relations defined by vectors $\vec{\mathbf{c}} = (c_1, c_2, \ldots, c_k)$ such that $c_1\geq c_2\geq\cdots \geq c_k$ and where $c_k = 1$. Under these conditions, we prove that every integer vector in $\mathbb{Z}^{k-1}$ admits a unique $\vec{\mathbf{c}}$-satisfying representation ($\vec{\mathbf{c}}$-SR) as a linear combination of vectors, $(\vec{\mathbf{X}}_n)_{n\in \mathbb{Z}}$ defined for every $n\in \mathbb{Z}$ by initially by zero and standard unit vectors and then the recursion $$\vec{\mathbf{X}}_{n} := c_1\vec{\mathbf{X}}_{n -1} + c_2\vec{\mathbf{X}}_{n - 2} + \cdots + c_k\vec{\mathbf{X}}_{n-k}.$$ To establish this, we introduce carrying and borrowing operations that use the defining recursion to transform any $\vec{\mathbf{c}}$ representation into a $\vec{\mathbf{c}}$-SR while preserving the underlying vector. Then, by establishing bijections with properties of scalar Positive Linear Recurrence Sequences (PLRS), we prove that these multidimensional decompositions inherit various properties, such as the number of summands exhibits Gaussian behavior and summand minimality of $\vec{\mathbf{c}}$-SRs over all all $\vec{\mathbf{c}}$-representations.

Figures (4)

• Figure 4.1: Regions $D_1$, …, $D_6$ for $\vec{\mathbf{c}}=(2,1,1)$. The black square indicates $\vec{\mathbf{0}} \in \mathbb{Z}^2$.
• Figure 4.2: Region $D_{10}$ and regions $R_1$, …, $R_{10}$ for $\vec{\mathbf{c}}=(2,1,1)$ respectively.
• Figure 4.3: Regions $D_1$, …, $D_6$ for $\vec{\mathbf{c}}=(1,2,1)$. The black square indicates $\vec{\mathbf{0}} \in \mathbb{Z}^2$.
• Figure 4.4: Region $D_{10}$ and regions $R_1$, …, $R_{10}$ for $\vec{\mathbf{c}}=(1,2,1)$ respectively.

Theorems & Definitions (33)

• Definition 1.1
• Remark 1.2
• Example 1.3
• Definition 1.4
• Remark 1.5
• Example 1.6
• Definition 1.7
• Theorem 1.8
• Definition 1.9
• Theorem 1.10