On the regularity of $\{\lfloor {\log }_b(\alpha n+\beta )\rfloor {\}}_{n\ge 0}$

Jiemeng Zhang, Yingjun Guo, Zhixiong Wen

Let $\alpha ,\beta$ be real numbers and $b\ge 2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloor \alpha n+\beta \rfloor {\}}_{n\ge 0}$ is $b$-regular if and only if $\alpha$ is rational.…