Boundary phenomena for slow recurrences:finite and infinite critical sets
Abstract
For a second-order recurrence with positive integral coefficients, the slow length of an integer is the largest index at which it can occur when the initial terms range over positive integers. Spiro conjectured that, among a finite set of relatively prime recurrence pairs, a pair is slowest infinitely often precisely when its growth constant is strictly less than the square of the smallest growth constant. We study the equality case. For Tm={(m−1,m),(m2−1,m2)}, the larger-growth pair attains the maximal slow length, with ties allowed, on Θm(log x) integers up to x whenever m≥3, whereas for m=2 its complete set of slowest integers is {1,2,25}. We give certificates for the infinite examples and prove a bounded-seed theorem that limits the possible initial pairs at each critical index.
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BibTeX
@misc{aliabadi2026boundary,
title = {{Boundary phenomena for slow recurrences:finite and infinite critical sets}},
author = {Mohsen Aliabadi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05424},
eprint = {2611.05424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05424},
}
RIS
TY - UNPB
TI - Boundary phenomena for slow recurrences:finite and infinite critical sets
AU - Mohsen Aliabadi
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.05424
UR - https://arxiv.org/abs/2611.05424
AB - For a second-order recurrence with positive integral coefficients, the slow length of an integer is the largest index at which it can occur when the initial terms range over positive integers. Spiro conjectured that, among a finite set of relatively prime recurrence pairs, a pair is slowest infinitely often precisely when its growth constant is strictly less than the square of the smallest growth constant. We study the equality case. For T_m={(m−1,m),(m^2−1,m^2)}, the larger-growth pair attains the maximal slow length, with ties allowed, on Θ_m(log x) integers up to x whenever m≥3, whereas for m=2 its complete set of slowest integers is {1,2,25}. We give certificates for the infinite examples and prove a bounded-seed theorem that limits the possible initial pairs at each critical index.
ER -