arXiv subjects
Publications and source records attributed to Yingjun Guo.
In this paper, we prove that a class of regular sequences can be viewed as projections of fixed points of uniform morphisms on a countable alphabet, and also can be generated by countable states automata. Moreover, we prove that the regularity of some regular sequences is invariant under some codings.
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.
In this paper, improving the method of Allouche \emph{et al.} \cite{APWW98}, we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence module 2 is periodic with period 10 which answers Coon's conjecture \cite{CV12}. Then we extend Bugeaud's method \cite{Bugeaud11} to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.