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59
On Finite Pseudorandom Binary Sequences, IV. (The Liouville Function, II)
, 2000
"... this paper, we shall use the following notations: p i for the ith prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) ( ..."
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Cited by 34 (10 self)
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this paper, we shall use the following notations: p i for the ith prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) (this is the Liouville function) and (n) = ( 1) !(n) so that (n) is completely multiplicative and (n) is multiplicative, and let LN = f(1); (2); : : : ; (N)g GN = f(1); (2); : : : ; (N)g: For y 1 let y (n) and y (n) denote the multiplicative functions de ned by y (p ( 1) (= (p +1 for p > y y (p 1 (= (p +1 for p > y; respectively, and write LN (y) = f y (1); y (2); : : : ; y (N)g GN (y) = f y (1); y (2); : : : ; y (N)g: Research partially supported by Hungarian National Foundation for Scienti c Research, Grant No. T017433 MKM fund FKFP0139/1997 and by FrenchHungarian APAPEOMFB exchange program F5 /97
On the complexity of algebraic numbers I. Expansions in integer bases
, 2005
"... Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. O ..."
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Cited by 33 (21 self)
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Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
Symbolic Dynamics and Finite Automata
, 1999
"... this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund. ..."
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Cited by 26 (9 self)
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this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund.
Palindromic continued fractions
 Ann. Inst. Fourier
"... On the complexity of algebraic numbers, II. ..."
Repetitive Delone sets and quasicrystals
, 1999
"... This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) ..."
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Cited by 18 (0 self)
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This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) of radius T being finite for all T. A Delone set X of finite type is repetitive if there is a function MX(T) such that every closed ball of radius MX(T)+T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R naction. There is a lower bound for MX(T) in terms of NX(T), namely MX(T) ≥ c(NX(T)) 1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T) purely in terms of NX(T). The complexity of a repetitive Delone set of finite type is measured by the growth rate of its repetitivity function MX(T). For example, the function MX(T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T) = O(T) as T → ∞ and is densely repetitive if MX(T) = O(NX(T)) 1/n as T → ∞. We show that linearly repetitive sets
A characterization of Sturmian morphisms
 Mathematical Foundations of Computer Science 1993, Lecture Notes in Computer Science
, 1993
"... Abstract. A morphism is called Sturmian if it preserves all Sturmian (infinite) words. It is weakly Sturmian if it preserves at least one Sturmian word. We prove that a morphism is Sturmian if and only if it keeps the word ba2ba2baba 2 bab balanced. As a consequence, weakly Sturmian morphisms are St ..."
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Cited by 16 (1 self)
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Abstract. A morphism is called Sturmian if it preserves all Sturmian (infinite) words. It is weakly Sturmian if it preserves at least one Sturmian word. We prove that a morphism is Sturmian if and only if it keeps the word ba2ba2baba 2 bab balanced. As a consequence, weakly Sturmian morphisms are Sturmian. An application to infinite words associated to irrational numbers is given. 1
There Are Ternary Circular SquareFree Words of Length n for n ≥ 18
, 2002
"... There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson. ..."
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Cited by 14 (1 self)
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There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson.
Strictly ergodic subshifts and associated operators
, 2005
"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."
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Cited by 13 (8 self)
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We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.
A Geometric Proof of the Enumeration Formula for Sturmian Words
, 1992
"... The number of factors of length m of Sturmian words is known to be 1 + P m 1 (m \Gamma i + 1)OE(i): We give a geometric proof of this formula, based on duality and on Euler's relation for planar graphs. ..."
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Cited by 12 (2 self)
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The number of factors of length m of Sturmian words is known to be 1 + P m 1 (m \Gamma i + 1)OE(i): We give a geometric proof of this formula, based on duality and on Euler's relation for planar graphs.