Kepler Sets of Second-Order Linear Recurrence Sequences Over $\mathbb{Q}_p$

TL;DR

This paper investigates the behavior of ratios in second-order linear recurrence sequences over p-adic numbers, focusing on their limit points within the p-adic field.

Contribution

It introduces a detailed analysis of the limit points of ratios in p-adic recurrence sequences, a topic not extensively explored before.

Findings

Characterization of limit points in p-adic recurrence ratios

Conditions under which ratios converge in $\, ext{Q}_p$

Insights into the dynamics of p-adic linear recurrences

Abstract

Let $(a_n)_{n=0}^\infty$ be a second-order linear recurrence sequence with constant coefficients over the field of $p$-adic numbers $\mathbb{Q}_p$. We study the set of limit points of the sequence of consecutive ratios $(a_{n+1}/a_{n})_{n=0}^\infty$ in $\mathbb{Q}_p$.