scholarly journals Higher diophantine approximation exponents and continued fraction symmetries for function fields

2011 ◽  
Vol 139 (01) ◽  
pp. 11-11 ◽  
Author(s):  
Dinesh S. Thakur
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Function Fields

scholarly journals Dirichlet's Theorem in Function Fields

2017 ◽  
Vol 69 (3) ◽  
pp. 532-547 ◽  
Author(s):  
Arijit Ganguly ◽  
Anish Ghosh
Keyword(s):  
Diophantine Approximation ◽  
Function Fields ◽  
Dirichlet’S Theorem ◽  
Dirichlet's Theorem

AbstractWe studymetric Diophantine approximation for function fields, specifically, the problem of improving Dirichlet's theorem in Diophantine approximation.

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

scholarly journals The continuous Diophantine approximation mapping of Szekeres

1995 ◽  
Vol 59 (2) ◽  
pp. 148-172 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Andrew D. Pollington
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Unit Square ◽  
Alternative Characterization ◽  
Continuous Analogue

AbstractSzekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

scholarly journals Non-periodic continued fractions in hyperelliptic function fields

2001 ◽  
Vol 64 (2) ◽  
pp. 331-343 ◽  
Author(s):  
Alfred J. van der Poorten
Keyword(s):  
Exponential Growth ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Function Fields ◽  
Square Root ◽  
90Th Birthday ◽  
Continued Fraction Expansion ◽  
Generic Polynomial ◽  
Partial Quotients ◽  
Hyperelliptic Function

Dedicated to George Szekeres on his 90th birthdayWe discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

scholarly journals Farey Sequences for Thin Groups

2021 ◽  
Author(s):  
Christopher Lutsko
Keyword(s):  
Explicit Formula ◽  
Diophantine Approximation ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Gauss Map ◽  
Gauss Measure ◽  
Farey Sequence ◽  
Farey Sequences ◽  
Partial Quotients

Abstract The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to study interesting number theoretic sequences—for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

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