Gaps for PLRS with zero coefficients
Abstract
Zeckendorf proved that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Among its extensions, we focus on Positive Linear Recurrence Sequences (PLRS): fixed depth recurrences with non-negative integral coefficients whose first and last terms are positive: Gn+1 = c1 Gn + … + cL Gn−L+1 with ci non-negative integers and c1 cL ≥ 1. We again find every integer can be written uniquely as an appropriate sum of terms of the sequence, and we can study the distribution of gaps between summands in decompositions. Bower et al. (2015) proved that if each ci ≥ 1 then the distribution of gaps in the generalized Zeckendorf decompositions of integers in [Gn, Gn+1) converges to a geometric decay for gaps at least two. What if some of the ci's are allowed to be zero? To date only the Kangaroo case, where there is a fixed g such that ci = 1 if i ≡ 1 bmod g and 0 otherwise, has been done and again geometric behavior emerges. Ben-Ari and Miller (2016) developed a probabilistic framework that also produce similar results, but their approach requires more advanced machinery. One of the goals of this paper is to show that the elementary methods used by Bower et al. (2015) can be adapted to this setting as well. We analyze the case Hn+1 = Hn + Hn−p + Hn−p−q. For gaps at least max(p,q) we again prove geometric decay, with parameter the largest root of the characteristic polynomial associated with the recurrence. For the finitely many smaller gaps, under the restriction p<q+2, we prove that the limiting probabilities exist and can be computed using a transfer-matrix method, yielding results that depend on p,q, and the gap size g. The case p ≥ q+2 is not addressed by our small-gap argument.
Related
- Voevodsky motives and motives with modulus with finite coefficients in characteristic zero
- Gaps theorems for linear forms with cubic coefficients
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- Zero sets in spaces of analytic functions with p-summable Taylor coefficients
BibTeX
@misc{chen2026gaps,
title = {{Gaps for PLRS with zero coefficients}},
author = {Yinxi Chen and Zongyun Chen and Brian Hii and Steven J. Miller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23991},
eprint = {2610.23991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23991},
}
RIS
TY - UNPB
TI - Gaps for PLRS with zero coefficients
AU - Yinxi Chen
AU - Zongyun Chen
AU - Brian Hii
AU - Steven J. Miller
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.23991
UR - https://arxiv.org/abs/2610.23991
AB - Zeckendorf proved that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Among its extensions, we focus on Positive Linear Recurrence Sequences (PLRS): fixed depth recurrences with non-negative integral coefficients whose first and last terms are positive: G_n+1 = c_1 G_n + … + c_L G_n−L+1 with c_i non-negative integers and c_1 c_L ≥ 1. We again find every integer can be written uniquely as an appropriate sum of terms of the sequence, and we can study the distribution of gaps between summands in decompositions. Bower et al. (2015) proved that if each c_i ≥ 1 then the distribution of gaps in the generalized Zeckendorf decompositions of integers in [G_n, G_n+1) converges to a geometric decay for gaps at least two. What if some of the c_i's are allowed to be zero? To date only the Kangaroo case, where there is a fixed g such that c_i = 1 if i ≡ 1 bmod g and 0 otherwise, has been done and again geometric behavior emerges. Ben-Ari and Miller (2016) developed a probabilistic framework that also produce similar results, but their approach requires more advanced machinery. One of the goals of this paper is to show that the elementary methods used by Bower et al. (2015) can be adapted to this setting as well. We analyze the case H_n+1 = H_n + H_n−p + H_n−p−q. For gaps at least max(p,q) we again prove geometric decay, with parameter the largest root of the characteristic polynomial associated with the recurrence. For the finitely many smaller gaps, under the restriction p<q+2, we prove that the limiting probabilities exist and can be computed using a transfer-matrix method, yielding results that depend on p,q, and the gap size g. The case p ≥ q+2 is not addressed by our small-gap argument.
ER -