An analogue of a theorem of Szüsz for formal Laurent series over finite fields.
The author proves an analogue to a generalization of the classical Gauss-Kuzmin theorem by the reviewer to formal Laurent series. My result is as follows. Let \(\alpha\in (0,1)\), \(\alpha= [a_0; a_1,\dots]\), that is, \([a_0; a_1,\dots]\) is the regular continued fraction expansion of the irrational number \(\alpha\). Denote (1) \(\frac{A_n}{B_n}= [a_0; a_1,\dots, a_n]\) in its irreducible form, (2) \(t_n= [0; a_n,\dots]\), further \(m(r,b,c,x)\) the Lebesgue-measure of the \(\alpha\)'s in \((0,1)\) satisfying \(B_n\equiv b\pmod r\), \(B_{n-1}\equiv c\pmod r\) and \(t_n< x\). Then we have \(m_n(r,b,c,x)= K\frac{\log(1+x)} {\log 2} (1+\delta^n)\), where \(K= \prod_{p\mid r} (1-\frac{1}{p^2})\) and \(\delta< 1\) if \((b,c,r)= 1\) and 0 otherwise. The mentioned analogue is that given a finite field \(F\) with \(q= p^m\) elements, one considers the set of formal Laurent series \(\sum a_kT^k\), where \(a_k\in F\). For \(\alpha= \sum_{k\leq n,a_n\neq 0} a_kT^k\) introduce the valuation (3) \(|\alpha|= q^n\). As in the classical case, for any \(\alpha\) there is a continued fraction representation (4) \(\alpha= \sum_{k\leq n} a_kT^k= [C_0; C_1,\dots]\), where the \(C\)'s are polynomials over \(F\). For \(r\geq 1\) we have \(|[0; C_r, C_{r+1},\dots]|< 1\); here \(| \dots |\) denotes the valuation (3). The author determines the probability \(m_j(R,S,T,x)=: p(\alpha\in F\), \(| \alpha |< 1\), \(B_{j-1}\equiv S\pmod R\), \(B_j\equiv T\pmod R\), \(t_j< x)\), where \(r\), \(S\) and \(T\) are fixed polynomials over \(F\) and \(B_{j-1}\), \(B_j\) are the denominators of the convergents of (4). The main result of the paper is Theorem 1. We have \[ m_j(R,S,T,x)= Kq^{[\log_qx]} (1+ O(\rho^j)) \text{ for }(S,T,R)=1,\;0\text{ otherwise}. \] Here \(\rho< 1\) and \(K= (| R|^2 \prod_{P\mid R} (1- \frac{1} {| P|^2}))^{-1}\). Further theorems of the paper establish the weak dependence of the random variables\break \((0; C_n, C_{n+1},\dots)\) \((n=1,2,\dots)\) and also give applications which are analogous to the classical case.
scientific article; zbMATH DE number 1957103