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 $T_m=\{(m-1,m),(m^2-1,m^2)\}$, the larger-growth pair attains the maximal slow length, with ties allowed, on $Θ_m(\log x)$ integers up to $x$ whenever $m\ge3$, 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.