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83
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 32 (20 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.
Polynomial versus exponential growth in repetitionfree binary words
 To appear, J. Combinatorial Theory Ser. A
, 2003
"... It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of le ..."
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Cited by 18 (4 self)
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It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of length n that avoid 7 3powers, but there are exponentially many binary words of length n that avoid 7+ 3powers. This answers an open question of Kobayashi from 1986. 1
A Structural Approach to Reversible Computation
 Theoretical Computer Science
, 2001
"... Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop ..."
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Cited by 18 (3 self)
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Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop a more structural approach. We show how highlevel functional programs can be mapped compositionally (i.e. in a syntaxdirected fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1
Palindromic continued fractions
 Ann. Inst. Fourier
"... On the complexity of algebraic numbers, II. ..."
Productivity of Stream Definitions
, 2008
"... We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called ‘productive’ if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas prod ..."
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Cited by 13 (3 self)
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We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called ‘productive’ if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for ‘pure’ stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called ‘pebbles’, in a finite ‘pebbleflow net(work)’. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets.
Diophantine properties of real numbers generated by finite automata
 Compos. Math
"... Abstract. We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1. ..."
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Cited by 10 (3 self)
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Abstract. We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1.
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 10 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Avoiding large squares in infinite binary words
 Turku Centre for Computer Science, TUCS General
, 2003
"... We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with y  ≥ 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary wor ..."
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Cited by 10 (3 self)
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We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with y  ≥ 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 0 2, 1 2, and (01) 2; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answer an open question of Prodinger and Urbanek from 1979 by demonstrating the existence of two infinite binary words, each avoiding arbitrarily large squares, such that their perfect shuffle has arbitrarily large squares. 1
A generalised Skolem–Mahler–Lech theorem for affine varieties
 J. London Math. Soc
"... The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m � 0 and a finite set. We prove that if X is a subvariety of a ..."
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Cited by 10 (0 self)
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The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m � 0 and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y,andσ is an automorphism of Y, then the set of m such that σ m (q) lies in X is a union of a finite number of complete doublyinfinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem. 1.
A partial order on ×2invariant measures
, 2006
"... Dedicated to the memory of Bill Parry Abstract. We introduce a partial order on the set of ×2invariant probability measures, describing relative dispersion, and show that the minimal elements for this order are the Sturmian measures of Morse & Hedlund. 1. ..."
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Cited by 8 (5 self)
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Dedicated to the memory of Bill Parry Abstract. We introduce a partial order on the set of ×2invariant probability measures, describing relative dispersion, and show that the minimal elements for this order are the Sturmian measures of Morse & Hedlund. 1.