Run-hierarchical structure of digital lines with irrational slopes in terms of continued fractions and the Gauss map

2009 ◽  
Vol 42 (10) ◽  
pp. 2247-2254 ◽  
Author(s):  
Hanna Uscka-Wehlou
Keyword(s):  
Hierarchical Structure ◽  
Continued Fractions ◽  
Gauss Map ◽  
Digital Lines

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.

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 Automatic detection of the optimal ejecting direction based on a discrete Gauss map

2014 ◽  
Vol 1 (1) ◽  
pp. 48-54 ◽  
Author(s):  
Masatomo Inui ◽  
Hidekazu Kamei ◽  
Nobuyuki Umezu
Keyword(s):  
Hierarchical Structure ◽  
Structural Strength ◽  
Gauss Map ◽  
Cad Model ◽  
Dome A ◽  
Distribution Method ◽  
Point Distribution ◽  
Optimal Direction ◽  
Plastic Parts ◽  
Geodesic Dome

Abstract In this paper, the authors propose a system for assisting mold designers of plastic parts. With a CAD model of a part, the system automatically determines the optimal ejecting direction of the part with minimum undercuts. Since plastic parts are generally very thin, many rib features are placed on the inner side of the part to give sufficient structural strength. Our system extracts the rib features from the CAD model of the part, and determines the possible ejecting directions based on the geometric properties of the features. The system then selects the optimal direction with minimum undercuts. Possible ejecting directions are represented as discrete points on a Gauss map. Our new point distribution method for the Gauss map is based on the concept of the architectural geodesic dome. A hierarchical structure is also introduced in the point distribution, with a higher level “rough” Gauss map with rather sparse point distribution and another lower level “fine” Gauss map with much denser point distribution. A system is implemented and computational experiments are performed. Our system requires less than 10 seconds to determine the optimal ejecting direction of a CAD model with more than 1 million polygons.

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 $[0,1]$.

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 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

A generalization of the Gauss map and some classical theorems on continued fractions

Nonlinearity ◽  
2004 ◽  
Vol 18 (2) ◽  
pp. 505-525 ◽  
Author(s):  
Simone D Cruz ◽  
Luiz Fernando C da Rocha
Keyword(s):  
Continued Fractions ◽  
Gauss Map

scholarly journals A noncommutative Gauss map

2011 ◽  
Vol 108 (2) ◽  
pp. 233 ◽  
Author(s):  
Caleb Eckhardt
Keyword(s):  
Continued Fractions ◽  
Bernoulli Shift ◽  
Gauss Map ◽  
Natural Place ◽  
Noncommutative Setting

The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra $\mathfrak A$ considered separately by F. Boca and D. Mundici. The center of $\mathfrak A$ is isomorphic to $C[0,1]$, so we first consider the action of the Gauss map on $C[0,1]$ and then extend the map to $\mathfrak A$ and show that the extension inherits many desirable properties.

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