scholarly journals Constructive Diophantine approximation in generalized continued fraction Cantor sets

2018 ◽  
Vol 186 (3) ◽  
pp. 225-241
Author(s):  
Kalle Leppälä ◽  
Topi Törmä
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Cantor Sets

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 On a problem of K. Mahler: Diophantine approximation and Cantor sets

2006 ◽  
Vol 338 (1) ◽  
pp. 97-118 ◽  
Author(s):  
Jason Levesley ◽  
Cem Salp ◽  
Sanju L. Velani
Keyword(s):  
Diophantine Approximation ◽  
Cantor Sets

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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