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Authors: Mumtaz Hussain and Nikita Shulga
Volume 95, Issue C
Published: 25 June 2024 Publication History
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Abstract
We prove the Hausdorff dimension of various limsup sets over the field of formal power series. Typically, the upper bound is easier to establish by considering the natural covering of the underlying set. To establish the lower bound, we identify a suitable set that serves as a subset of several limsup sets by selecting appropriate values for the involved parameters. To be precise, given a fixed integer m and for all integers 0 ≤ i ≤ m − 1, let α i > 0 be a real number. Define the set F m ( α 0, …, α m − 1 ) = def { x ∈ I : deg A n + i = ⌊ n α i ⌋ + c i, 0 ≤ i ≤ m − 1 = def { x ∈ I for infinitely many n ∈ N }, where c i ∈ N are fixed, and the partial quotients A i ( x ) are polynomials of strictly positive degree. We then determine the Hausdorff dimension of this set which establishes an optimal lower bound for various sets of interest, including the key results in [6,8,9]. Some new applications of our theorem include the lower bound of the Hausdorff dimension of the formal power series analogues of the sets considered in [2,20]. We also prove their upper bounds to provide the comprehensive Hausdorff dimension analysis of these sets.
The main ingredient of the proof lies in the introduction of m probability measures consistently distributed over the Cantor-type subset of F m ( α 0, …, α m − 1 ).
References
[1]
E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I-II., Math. Z. 19 (1) (1924) 153–246.
[2]
A. Bakhtawar, P. Bos, M. Hussain, Hausdorff dimension of an exceptional set in the theory of continued fractions, Nonlinearity 33 (6) (2020) 2615–2639.
[3]
A. Bakhtawar, P. Bos, M. Hussain, The sets of Dirichlet non-improvable numbers versus well-approximable numbers, Ergod. Theory Dyn. Syst. 40 (12) (2020) 3217–3235.
[4]
V. Berthé, H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Expo. Math. 18 (4) (2000) 257–284.
[5]
P. Bos, M. Hussain, D. Simmons, The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers, Proc. Am. Math. Soc. 151 (5) (2023) 1823–1838.
[6]
Y. Feng, S. Shi, Y. Zhang, Metrical properties for the weighted sums of degrees of multiple partial quotients in continued fractions of Laurent series, Finite Fields Appl. 93 (2024).
[7]
M. Fuchs, On metric Diophantine approximation in the field of formal Laurent series, Finite Fields Appl. 8 (3) (2002) 343–368.
[8]
H. Hu, M. Hussain, Y. Yu, Metrical properties for continued fractions of formal Laurent series, Finite Fields Appl. 73 (2021).
[9]
X.-H. Hu, B.-W. Wang, J. Wu, Y.-L. Yu, Cantor sets determined by partial quotients of continued fractions of Laurent series, Finite Fields Appl. 14 (2) (2008) 417–437.
[10]
L. Huang, J. Wu, Uniformly non-improvable Dirichlet set via continued fractions, Proc. Am. Math. Soc. 147 (11) (2019) 4617–4624.
[11]
L. Huang, J. Wu, J. Xu, Metric properties of the product of consecutive partial quotients in continued fractions, Isr. J. Math. 238 (2) (2020) 901–943.
[12]
M. Hussain, D. Kleinbock, N. Wadleigh, B.-W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika 64 (2) (2018) 502–518.
[13]
M. Hussain, B. Li, N. Shulga, Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions, Discrete Contin. Dyn. Syst. 44 (1) (2024) 154–181.
[14]
Hussain, M.; Shulga, N. : Metrical properties of exponentially growing partial quotients. arXiv:2309.10529 preprint 2023.
[15]
T. Kim, W. Kim, Hausdorff measure of sets of Dirichlet non-improvable affine forms, Adv. Math. 403 (2022).
[16]
D. Kleinbock, A. Strömbergsson, S. Yu, A measure estimate in geometry of numbers and improvements to Dirichlet's theorem, Proc. Lond. Math. Soc. (3) 125 (4) (2022) 778–824.
[17]
D. Kleinbock, N. Wadleigh, A zero-one law for improvements to Dirichlet's theorem, Proc. Am. Math. Soc. 146 (5) (2018) 1833–1844.
[18]
B. Li, B. Wang, J. Xu, Hausdorff dimension of Dirichlet non-improvable set versus well-approximable set, Ergod. Theory Dyn. Syst. 43 (8) (2023) 2707–2731.
[19]
H. Niederreiter, Keystream sequences with a good linear complexity profile for every starting point, in: Advances in Cryptology—EUROCRYPT '89, in: Lecture Notes in Comput. Sci., vol. 434, Houthalen, 1989, Springer, Berlin, 1990, pp. 523–532.
[20]
B. Tan, C. Tian, B. Wang, The distribution of the large partial quotients in continued fraction expansions, Sci. China Math. 65 (2022).
[21]
B.-W. Wang, J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (5) (2008) 1319–1339.
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Published In
Finite Fields and Their Applications Volume 95, Issue C
Mar 2024
604 pages
ISSN:1071-5797
Issue’s Table of Contents
The Author(s).
Publisher
Elsevier Science Publishers B. V.
Netherlands
Publication History
Published: 25 June 2024
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- primary
Author Tags
- Hausdorff measure and dimension
- Limsup sets
- Metric continued fractions
- Formal Laurent series
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