The Shape of the Value Sets of Linear Recurrence Sequences

TL;DR

This paper characterizes the closure of the value sets of real linear recurrence sequences, showing they are unions of countable sets and intervals, and demonstrates any finite collection of intervals can be realized as such a closure.

Contribution

It provides a complete description of the topological structure of value sets of linear recurrence sequences and constructs sequences for any finite collection of intervals.

Findings

Closure of value set is union of countable set and finite intervals.

Any finite collection of closed intervals can be realized as a recurrence sequence's value set closure.

The structure of value sets is fully characterized in topological terms.

Abstract

We show that the closure of the value set of a real linear recurrence sequence is the union of a countable set and a finite collection of intervals. Conversely, any finite collection of closed intervals is the closure of the value set of some recurrence sequence.