arXiv subjects

Jiemeng Zhang

Publications and source records attributed to Jiemeng Zhang.

2 recordsLinked to original sources

arXiv2019

Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences

First, we show that the sum-free set generated by the period-doubling sequence is not $κ$-regular for any $κ\geq 2$. Next, we introduce a generalization of the period-doubling sequence, which we call the period-$k$-folding sequences. We show that the sum-free sets generated by the period-$k$-folding sequences also fail to be $κ$-regular for all $κ\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.

math.CO↗

arXiv2015

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

Let $α,β$ be real numbers and $b\geq2$ be an integer. Allouche and Shallit showed that the sequence $\{\lfloorαn+β\rfloor\}_{n\geq0}$ is $b$-regular if and only if $α$ is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence $\{\lfloor\log_b(αn+β)\rfloor\}_{n\geq0}$ is $b$-regular if and only if $α$ is rational. In particular, when $α=\sqrt{2},β=0$ 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.

cs.FL↗