Higher Diophantine approximation exponents and continued fraction symmetries for function fields. (English) Zbl 1222.11089
The author constructs some families of non quadratic algebraic Laurent series with continued fractions having a bounded partial quotients sequence and with the diophantine approximation exponent for approximation by quadratics being arbitrarily large. The author considers the cases where folding or negative reversal patterns of the relevant continued fractions become repeating or half-repeating in even or odd characteristic respectively.
The article is in the context of diophantine approximation of functions fields \(\mathbb F_q(t)\) where \(\mathbb F_q\) is a finite field of characteristic \(p\) containing \(q\) elements.
The author recalls the definitions of absolute and field height, higher diophantine exponents in \(\mathbb R\), (with the mention that these definitions are similar in \(\mathbb F_q(t)\)) and the classical notations of continued fractions expansions.
We give as an example two of the five theorems obtained. Let us note \(-X^-\)the tuple \((-a_n,\dots, -a_1)\) where \(X\) is the tuple \(X=(a_1,\dots,a_n)\) with \(a_i\in\mathbb F_q[t]\).
Repeat symmetry: Let us consider for arbitrary \(q\) and a tuple \(X_1\) and sequence \(y_n\) of partial quotients the continued fraction \[ \alpha=[0,X_1,y_1,X_1,y_2,X_2,\dots]=\lim[0,X_n],\;\;X_{n+1}=[X_n,y_n,X_n]. \] Let us denote the \(i\)th convergent of \(\alpha\) as \(p_i/q_i\), so that if \([0,X_n]\) has length \(m+1\) then \[ [0,X_n]=\frac{p_m}{q_m},\;[0,X_n,y_n]=\frac{p_{m+1}}{q_{m+1}},\;[0,X_n,y_n,X_n]=\frac{p_{2m+1}}{q_{2m+1}}. \] The author obtains:
Suppose that \(p=2\). Let \(\alpha\) as above with \(\deg y_n\) increasing and \(E:= 2+\lim \deg y_n/\deg y_m\). Then the exponent \(E_2(\alpha)\geq 3+1/(E-1)\).
Half-repeat symmetry: Let now \[ \alpha=[0,X_1,y_1,-X_1^-,y_2,X_1,-y_1,-X_1^-,y_3,X_1,\dots] \] with \[ [0,X_{n-1},y_{n-1},-X_{n-1}^-]=[0,X_n]=\frac{p_m}{q_m},\;[0,X_n,y_n]=\frac{p_{m+1}}{q_{m+1}}, \] and \[ [0,X_{n-1},y_{n-1},-X_{n-1}^-,y_n,X_{n-1}]=\frac{p_{m+1+(m-1)/2}}{q_{m+1+(m-1)/2}}. \] The author obtains:
Suppose \(p>2\) and \(\alpha\) as immediately above, with \(E:=2+\lim \deg\;y_n/\deg\;q_m\), then the exponent \(E_2(\alpha)\geq 2+1/(E-1)\).
These results are not completely situated in the historical context.
The article is in the context of diophantine approximation of functions fields \(\mathbb F_q(t)\) where \(\mathbb F_q\) is a finite field of characteristic \(p\) containing \(q\) elements.
The author recalls the definitions of absolute and field height, higher diophantine exponents in \(\mathbb R\), (with the mention that these definitions are similar in \(\mathbb F_q(t)\)) and the classical notations of continued fractions expansions.
We give as an example two of the five theorems obtained. Let us note \(-X^-\)the tuple \((-a_n,\dots, -a_1)\) where \(X\) is the tuple \(X=(a_1,\dots,a_n)\) with \(a_i\in\mathbb F_q[t]\).
Repeat symmetry: Let us consider for arbitrary \(q\) and a tuple \(X_1\) and sequence \(y_n\) of partial quotients the continued fraction \[ \alpha=[0,X_1,y_1,X_1,y_2,X_2,\dots]=\lim[0,X_n],\;\;X_{n+1}=[X_n,y_n,X_n]. \] Let us denote the \(i\)th convergent of \(\alpha\) as \(p_i/q_i\), so that if \([0,X_n]\) has length \(m+1\) then \[ [0,X_n]=\frac{p_m}{q_m},\;[0,X_n,y_n]=\frac{p_{m+1}}{q_{m+1}},\;[0,X_n,y_n,X_n]=\frac{p_{2m+1}}{q_{2m+1}}. \] The author obtains:
Suppose that \(p=2\). Let \(\alpha\) as above with \(\deg y_n\) increasing and \(E:= 2+\lim \deg y_n/\deg y_m\). Then the exponent \(E_2(\alpha)\geq 3+1/(E-1)\).
Half-repeat symmetry: Let now \[ \alpha=[0,X_1,y_1,-X_1^-,y_2,X_1,-y_1,-X_1^-,y_3,X_1,\dots] \] with \[ [0,X_{n-1},y_{n-1},-X_{n-1}^-]=[0,X_n]=\frac{p_m}{q_m},\;[0,X_n,y_n]=\frac{p_{m+1}}{q_{m+1}}, \] and \[ [0,X_{n-1},y_{n-1},-X_{n-1}^-,y_n,X_{n-1}]=\frac{p_{m+1+(m-1)/2}}{q_{m+1+(m-1)/2}}. \] The author obtains:
Suppose \(p>2\) and \(\alpha\) as immediately above, with \(E:=2+\lim \deg\;y_n/\deg\;q_m\), then the exponent \(E_2(\alpha)\geq 2+1/(E-1)\).
These results are not completely situated in the historical context.
Reviewer: Roland Quême (Brax)
MSC:
| 11J17 | Approximation by numbers from a fixed field |
| 11J70 | Continued fractions and generalizations |
| 11J93 | Transcendence theory of Drinfel’d and \(t\)-modules |
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