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18
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
, 2009
"... Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal ..."
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Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times.
FRACTAL TILES ASSOCIATED WITH SHIFT RADIX SYSTEMS
"... Abstract. Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the ddimensional real vector space. They generalize wellknown numeration systems such as betaexpansions, expansions with respect to rational bases, and canonical number systems. Betanu ..."
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Abstract. Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the ddimensional real vector space. They generalize wellknown numeration systems such as betaexpansions, expansions with respect to rational bases, and canonical number systems. Betanumeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the wellknown tiles associated with betaexpansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for betaexpansions with (nonunit) Pisot numbers as well as canonical number systems with (nonmonic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the ddimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be selfaffine (or graph directed selfaffine). 1.
On the cost and complexity of the successor function
 IN PROC. WORDS 2007
, 2009
"... For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if the ratio ..."
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Cited by 4 (3 self)
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For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if the ratio of the number of elements of length n +1overthenumber of elements of length n of the language has a limit β>1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems).
Diophantine approximation, Ostrowski numeration and the doublebase number system
 Discrete Math. Theor. Comput. Sci
, 2009
"... A partition of x> 0 of the form x = P i 2a i 3 b i with distinct parts is called a doublebase expansion of x. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest {2, 3}integer, i.e., a number of the form 2 a 3 b, less than or equal to ..."
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A partition of x> 0 of the form x = P i 2a i 3 b i with distinct parts is called a doublebase expansion of x. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest {2, 3}integer, i.e., a number of the form 2 a 3 b, less than or equal to x. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of doublebase expansions when only a few iterations of our algorithm are performed.
Rational selfaffine tiles
"... Dedicated to Professor Shigeki Akiyama on the occasion of his 50 th birthday Abstract. An integral selfaffine tile is the solution of a set equation AT = ⋃ d∈D (T +d), where A is an n × n integer matrix and D is a finite subset of Zn. In the recent decades, these objects and the induced tilings hav ..."
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Dedicated to Professor Shigeki Akiyama on the occasion of his 50 th birthday Abstract. An integral selfaffine tile is the solution of a set equation AT = ⋃ d∈D (T +d), where A is an n × n integer matrix and D is a finite subset of Zn. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Qn×n. We define rational selfaffine tiles as compact subsets of the open subring Rn × ∏ p Kp of the adèle ring AK, where the factors of the (finite) product are certain padic completions of a number field K that is defined in terms of the characteristic polynomial of A. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational selfaffine tile with Rn × ∏ p {0} ≃ Rn. Although these intersection tiles have a complicated structure and are no longer selfaffine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems. 1.
Rational base number systems for padic numbers
"... Abstract. This paper deals with rational base number systems for padic numbers. We mainly focus on the system proposed by Akiyama, Frougny, and Sakarovitch in 2008, but we also show that this system is in some sense isomorphic to other ones. We identify the numbers with finite and eventually period ..."
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Abstract. This paper deals with rational base number systems for padic numbers. We mainly focus on the system proposed by Akiyama, Frougny, and Sakarovitch in 2008, but we also show that this system is in some sense isomorphic to other ones. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given padic number. 1
Factor complexity of infinite words associated with nonsimple Parry
, 812
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FINITE AND PERIODIC ORBITS OF SHIFT RADIX SYSTEMS
"... Abstract. For r = (r0,..., rd−1) ∈ R d define the function τr: Z d → Zd, z = (z0,..., zd−1) 7 → (z1,..., zd−1,−⌊rz⌋), where rz is the scalar product of the vectors r and z. If each orbit of τr ends up at 0, we call τr a shift radix system. It is a wellknown fact that each orbit of τr ends up perio ..."
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Abstract. For r = (r0,..., rd−1) ∈ R d define the function τr: Z d → Zd, z = (z0,..., zd−1) 7 → (z1,..., zd−1,−⌊rz⌋), where rz is the scalar product of the vectors r and z. If each orbit of τr ends up at 0, we call τr a shift radix system. It is a wellknown fact that each orbit of τr ends up periodically if the polynomial td + rd−1t d−1+ · · ·+ r0 associated to r is contractive. On the other hand, whenever this polynomial has at least one root outside the unit circle, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of the mappings τr for vectors r associated to polynomials whose roots have modulus less than or equal to one with equality in at least one case. We show that for a large class of vectors r belonging to the above class the ultimate periodicity of the orbits of τr is equivalent to the fact that τs is a shift radix system or has another prescribed orbit structure for a certain parameter s related to r. These results are combined with new algorithmic results in order to characterize vectors r of the above class that give rise to ultimately periodic orbits of τr for each starting value. In particular, we work out the description of these vectors r for the case d = 3. This leads to sets which seem to have a very intricate structure. 1.
On sets of numbers rationally represented in a rational base number system
 Proceedings of CAI 2013
, 2013
"... Abstract. In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition of a strong combinatorial property on languages: th ..."
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Abstract. In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition of a strong combinatorial property on languages: the bounded left iteration property. It is both an unnatural property in usual formal language theory (as it contradicts any kind of pumping lemma) and an ideal fit to the languages defined through rational base number systems. 1