The minimal growth of a $k$-regular sequence

TL;DR

This paper establishes a lower growth bound for unbounded k-regular sequences, showing they grow at least logarithmically infinitely often, and addresses a question about sums of multiplicative automatic functions.

Contribution

It proves a minimal growth rate for unbounded k-regular sequences and answers a question on sums of multiplicative automatic functions.

Findings

Unbounded k-regular sequences grow at least logarithmically infinitely often.

There exists a constant c > 0 such that |f(n)| > c log n infinitely often.

The paper resolves a question about sums of completely multiplicative automatic functions.

Abstract

We determine a lower gap property for the growth of an unbounded \(\mathbb{Z}\)-valued \(k\)-regular sequence. In particular, if \(f:\mathbb{N}\to\mathbb{Z}\) is an unbounded \(k\)-regular sequence, we show that there is a constant \(c>0\) such that \(|f(n)|>c\log n\) infinitely often. We end our paper by answering a question of Borwein, Choi, and Coons on the sums of completely multiplicative automatic functions.