uu.seUppsala University Publications
Change search
ReferencesLink to record
Permanent link

Direct link
Two Equivalence Relations on Digital Lines with Irrational Slopes. A Continued Fraction Approach to Upper Mechanical Words
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. (Digital geometry and mathematical morphology)
2009 (English)In: Theoretical Computer Science, ISSN 0304-3975, Vol. 410, no 38--40, 3655-3669 p.Article in journal (Refereed) Published
Abstract [en]

We examine the influence of the elements of the continued fraction (CF) expansion of irrational positive a less than 1 on the construction of runs in the digitization of the positive half line y=ax or, equivalently, on the run-hierarchical structure of the upper mechanical word with slope a and intercept 0. Special attention is given to the CF elements equal to 1.We define two complementary equivalence relations on the set of slopes, based on their CF expansions. A new description of digital lines is presented; we show how to define a straight line or upper mechanical word by two sequences of positive integers fulfilling some extra conditions. These equivalence relations and this new description enable us to analyze the construction of digital lines and upper mechanical words. The analysis of suprema of equivalence classes under one of these relations leads to a result which involves Fibonacci numbers.

Place, publisher, year, edition, pages
Elsevier , 2009. Vol. 410, no 38--40, 3655-3669 p.
Keyword [en]
digital line, upper mechanical word, characteristic word, irrational slope, continued fraction, hierarchy, run, classes of digital lines, classes of words, Fibonacci numbers
National Category
URN: urn:nbn:se:uu:diva-105925DOI: 10.1016/j.tcs.2009.04.026ISI: 000269683400008OAI: oai:DiVA.org:uu-105925DiVA: diva2:222873
Ph.D. project
Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2010-07-07Bibliographically approved
In thesis
1. Digital lines, Sturmian words, and continued fractions
Open this publication in new window or tab >>Digital lines, Sturmian words, and continued fractions
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure.

Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction.

In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k.

In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines.

In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general.

In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2009. 59 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 65
digital geometry, digital line, hierarchy of runs, combinatorics on words, Sturmian word, upper mechanical word, characteristic word, irrational slope, continued fraction, Gauss map, fixed point
National Category
Discrete Mathematics
Research subject
urn:nbn:se:uu:diva-107274 (URN)978-91-506-2090-0 (ISBN)
Public defence
2009-09-25, Polhemsalen, Lägerhyddsvägen 1, Uppsala, Ångström Laboratory, 10:15 (English)
Available from: 2009-09-04 Created: 2009-08-03 Last updated: 2009-09-04Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text

Search in DiVA

By author/editor
Uscka-Wehlou, Hanna
By organisation
Department of Mathematics
In the same journal
Theoretical Computer Science

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 197 hits
ReferencesLink to record
Permanent link

Direct link