Metric properties and exceptional sets of β-expansions over formal Laurent series

2008 ◽  
Vol 155 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Bing Li ◽  
Jun Wu ◽  
Jian Xu
Keyword(s):  
Laurent Series ◽  
Formal Laurent Series ◽  
Metric Properties ◽  
Exceptional Sets

scholarly journals Exceptional sets of the Oppenheim expansions over the field of formal Laurent series

2016 ◽  
Vol 42 ◽  
pp. 253-268
Author(s):  
Mei-Ying Lü ◽  
Jia Liu ◽  
Zhen-Liang Zhang
Keyword(s):  
Laurent Series ◽  
Formal Laurent Series ◽  
Exceptional Sets

ON THE LARGEST DEGREE OF THE PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS OVER THE FIELD OF FORMAL LAURENT SERIES

2013 ◽  
Vol 09 (05) ◽  
pp. 1237-1247 ◽  
Author(s):  
LUMING SHEN ◽  
JIAN XU ◽  
HUIPING JING
Keyword(s):  
Continued Fraction ◽  
Laurent Series ◽  
Type Theorem ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Iterated Logarithm ◽  
Partial Quotients ◽  
Almost All ◽  
Continued Fraction Expansions

For x ∈ I, let [A1(x), A2(x), …] be the continued fraction expansions over the field of Laurent series, write Ln(x) ≔ max { deg A1(x), deg A2(x), …, deg An(x)}, which is called the largest degree of partial quotients. In this paper, we give an iterated logarithm type theorem for Ln(x), and by which, we get that for P-almost all x ∈ I, [Formula: see text]. Also the Hausdorff dimensions of the related exceptional sets are determined.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

scholarly journals Metrical properties for continued fractions of formal Laurent series

2021 ◽  
Vol 73 ◽  
pp. 101850
Author(s):  
Hui Hu ◽  
Mumtaz Hussain ◽  
Yueli Yu
Keyword(s):  
Continued Fractions ◽  
Laurent Series ◽  
Formal Laurent Series

Some remarks on formal power series and formal Laurent series

2017 ◽  
Vol 67 (3) ◽  
Author(s):  
Dariusz Bugajewski ◽  
Xiao-Xiong Gan
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Formal Power

AbstractIn this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.:

scholarly journals Betti realization of varieties defined by formal Laurent series

2021 ◽  
Vol 25 (4) ◽  
pp. 1919-1978
Author(s):  
Piotr Achinger ◽  
Mattia Talpo
Keyword(s):  
Laurent Series ◽  
Formal Laurent Series
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