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.

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 the sets of $A,S,U$-numbers is nonempty. On the other hand, the existence of $T$-numbers has been an open problem. In this paper, we prove that they exist.

In this paper, we study the properties of 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 $(\xi _j)_j$ such that $\deg \xi _j =p^{k_j}+1$ and $$ w_1(\xi _j)=w_1 ^{∗}(\xi _j)=\ldots =w_n(\xi _j)=w_n ^{∗}(\xi _j)=w $$ for any $j \geq 1$. For each $n \geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{∗}(\xi )$ are different.