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Let $\mathbb{F}_q$ be a finite field with q elements and $\mathbb{F}_q((X^{-1}))$ be the field of all formal Laurent series with coefficients lying in $\mathbb{F}_q$. This paper concerns with the size of the set of points $x{\in}\mathbb{F}_q((X^{-1}))$ with their partial quotients $A_n(x)$ both lying in a given subset $\mathbb{B}$ of polynomials in $\mathbb{F}_q[X]$ ($\mathbb{F}_q[X]$ denotes the ring of polynomials with coefficients in $\mathbb{F}_q$) and deg $A_n(x)$ tends to infinity at least with some given speed. Write $E_{\mathbb{B}}=\{x:A_n(x){\in}\mathbb{B},\;deg\;A_n(x){\rightarrow}{\infty}\;as\;n{\rightarrow}{\infty}\}$. It was shown in [8] that the Hausdorff dimension of $E_{\mathbb{B}}$ is inf{$s:{\sum}_{b{\in}\mathbb{B}}(q^{-2\;deg\;b})^s$ < ${\infty}$}. In this note, we will show that the above result is sharp. Moreover, we also attempt to give conditions under which the above dimensional formula still valid if we require the given speed of deg $A_n(x)$ tends to infinity.
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참고문헌 (20)
참고문헌 신청
A. D. Pollington / 2000 / On a problem in simultaneous Diophantine approxi- mation: Littlewood's conjecture / Acta Math 185 (2) : 287 ~ 306
A. Lasjaunias / 2000 / A survey of Diophantine approximation in fields of power series / Monatsh. Math 130 (3) : 211 ~ 229
B. W. Wang / 2008 / A problem of Hirst on continued fractions with sequences of partial quotients / Bull. Lond. Math. Soc 40 (1) : 18 ~ 22
D. H. Kim / 2007 / The shrinking target property of irrational rotations / Nonlinearity 20 (7) : 1637 ~ 1643
E. Artin / 1924 / Quadratische Korper im Gebiete der hoheren Kongruenzen. I-II / Math. Z 19 (1) : 153 ~ 246
A. Lasjaunias / 2000 / A survey of Diophantine approximation in fields of power series / Monatsh. Math 130 (3) : 211 ~ 229
B. W. Wang / 2008 / A problem of Hirst on continued fractions with sequences of partial quotients / Bull. Lond. Math. Soc 40 (1) : 18 ~ 22
D. H. Kim / 2007 / The shrinking target property of irrational rotations / Nonlinearity 20 (7) : 1637 ~ 1643
E. Artin / 1924 / Quadratische Korper im Gebiete der hoheren Kongruenzen. I-II / Math. Z 19 (1) : 153 ~ 246
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