2025/10/08 by Jia Xing Cheng, Cheng, Jiarui, Steven J. Miller +7
Computer Science · Mathematics · #11A67 #11B34 #11B39 #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.07237
openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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 c = (c1, c2, …, ck) such that c1≥ c2≥⋯ ≥ ck and where ck = 1. Under these conditions, we prove that every integer vector in ℤk-1 admits a unique c-satisfying representation (c-SR) as a linear combination of vectors, (Xn)n∈ ℤ defined for every n∈ ℤ by initially by zero and standard unit vectors and then the recursion Xn := c1Xn -1 + c2Xn - 2 + ⋯ + ckXn-k. To establish this, we introduce carrying and borrowing operations that use the defining recursion to transform any c representation into a 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 c-SRs over all all c-representations.