arXiv subjects
Publications and source records attributed to Tomohiro Ooto.
We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.
Schleischitz [arXiv:1701.01129] determined exponents of best approximations to a strong Liouville number by integer polynomials and algebraic numbers of precribed degree. In this note, we show that we cannot extend his result to arbitrary Liouville numbers.
As an analogue of Mahler's classification for real numbers, Bundschuh introduced a classification for Laurent series over a finite field, divided into $A,S,T,U$-numbers.It is known that each of $A,S,U$-numbers is nonempty.On the other hand, the existence of $T$-numbers is open.In this paper, we give an affirmative answer to the problem.
In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j\geq 1}$ and a sequence of algebraic Laurent series $(ξ_j)_{j\geq 1}$ such that deg $ξ_j=p^{k_j}+1$ and \begin{equation} w_1(ξ_j)=w_1 ^{*}(ξ_j)=\ldots =w_n(ξ_j)=w_n ^{*}(ξ_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $ξ$ for which $w_n(ξ)$ and $w_n^{*}(ξ)$ are different.
We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.
In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2 ^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.
In this paper, we study a relation between digits of $p$-adic numbers and Mahler's classification. We show that an irrational $p$-adic number whose digits are automatic, primitive morphic, or Sturmian is an $S$-, $T$-, or $U_1$-number in the sense of Mahler's classification. Furthermore, we give an algebraic independence criterion for $p$-adic numbers whose digits are Sturmian.