On the regularity of $\{\lfloor\log_b(\alpha n+\beta)\rfloor\}_{n\geq0}$

TL;DR

This paper proves that the sequence of floor logarithms of a linear function is $b$-regular if and only if the coefficient is rational, extending previous results about linear sequences and addressing an open question.

Contribution

It establishes a base-independent regularity criterion for logarithmic sequences, showing they are $b$-regular only when the coefficient is rational, and resolves an open problem about the sequence involving $rac{1}{2}+ ext{log}_2 n$.

Findings

Sequences with irrational $eta$ are not $b$-regular.

Sequences with rational $eta$ are $b$-regular.

The specific sequence $ig floor rac{1}{2}+ ext{log}_2 n ig floor$ is not 2-regular.

Abstract

Let $\alpha,\beta$ be real numbers and $b\geq2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloor\alpha n+\beta\rfloor\}_{n\geq0}$ is $b$-regular if and only if $\alpha$ is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence $\{\lfloor\log_b(\alpha n+\beta)\rfloor\}_{n\geq0}$ is $b$-regular if and only if $\alpha$ is rational. In particular, when $\alpha=\sqrt{2},\beta=0$ and $b=2$, we answer the question of Allouche and Shallit that the sequence $\{\lfloor\frac{1}{2}+\log_2n\rfloor\}_{n\geq0}$ is not $2$-regular, which has been proved by Bell, Moshe and Rowland respectively.