scholarly journals Decreasing height along continued fractions

2018 ◽  
Vol 40 (3) ◽  
pp. 763-788
Author(s):  
GIOVANNI PANTI
Keyword(s):  
Continued Fractions ◽  
Number Fields ◽  
Gauss Map ◽  
Projective Line ◽  
Triangle Groups ◽  
Quadratic Invariant ◽  
Weil Height ◽  
Symbolic Sequences ◽  
Commensurability Class ◽  
Quadratic Integer

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

A generalization of the Jarník–Besicovitch theorem by continued fractions

2015 ◽  
Vol 36 (4) ◽  
pp. 1278-1306 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Gauss Map ◽  
Partial Quotient ◽  
Image Position ◽  
Positive Functions ◽  
Besicovitch Sets

We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where $T$ is the Gauss map and $a_{n}(x)$ is the $n$th partial quotient of $x$. In this paper, we consider the size of the generalized Jarník–Besicovitch set $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where ${\it\tau}(x)$ and $f(x)$ are positive functions defined on $[0,1]$.

scholarly journals Generalized continued fractions and orbits under the action of Hecke triangle groups

2008 ◽  
Vol 134 (4) ◽  
pp. 337-348 ◽  
Author(s):  
Elise Hanson ◽  
Adam Merberg ◽  
Christopher Towse ◽  
Elena Yudovina
Keyword(s):  
Continued Fractions ◽  
Triangle Groups ◽  
Generalized Continued Fractions

On the dual of Rauzy induction

2016 ◽  
Vol 37 (5) ◽  
pp. 1492-1536 ◽  
Author(s):  
KAE INOUE ◽  
HITOSHI NAKADA
Keyword(s):  
Continued Fractions ◽  
Natural Extension ◽  
Gauss Map ◽  
Interval Exchange ◽  
Dual Relationship ◽  
Rauzy Induction ◽  
Interval Exchange Maps

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity18 (2005), 505–525]  introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

scholarly journals Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jordan Schettler
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Continued Fraction ◽  
Number Fields ◽  
Quadratic Number ◽  
Continued Fraction Expansion ◽  
Quadratic Number Fields ◽  
Analogous Formula ◽  
Partial Quotients ◽  
Imaginary Quadratic Number

Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.

scholarly journals Renewal-type limit theorem for the Gauss map and continued fractions

2008 ◽  
Vol 28 (2) ◽  
pp. 643-655 ◽  
Author(s):  
YAKOV G. SINAI ◽  
CORINNA ULCIGRAI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Natural Extension ◽  
Gauss Map ◽  
Limiting Distribution ◽  
Special Flow

AbstractIn this paper we prove a renewal-type limit theorem. Given $\alpha \in (0,1)\backslash \mathbb {Q}$ and R>0, let qnR be the first denominator of the convergents of α which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.

scholarly journals Renewal-type limit theorem for the Gauss map and continued fractions

Selecta ◽  
2010 ◽  
pp. 127-140
Author(s):  
Yakov G. Sinai ◽  
Corinna Ulcigrai
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Gauss Map

scholarly journals CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 21-42 ◽  
Author(s):  
KARIANE CALTA ◽  
THOMAS A. SCHMIDT
Keyword(s):  
Continued Fractions ◽  
Fuchsian Group ◽  
Modular Group ◽  
Continued Fraction ◽  
Piecewise Linear ◽  
Triangle Groups ◽  
Interval Map ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Natural Extensions

AbstractWe give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

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