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57
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 92 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
Ergodic properties of Erdős measure, the entropy of the goldenshift and related problems
 MATH
, 1998
"... We define a twosided analog of Erdös measure on the space of twosided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T² which is the induced ..."
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Cited by 46 (19 self)
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We define a twosided analog of Erdös measure on the space of twosided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T² which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdős measure on the interval, its entropy in the sense of GarsiaAlexanderZagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 40 (5 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
Random Series In Powers Of Algebraic Integers: Hausdorff Dimension Of The Limit Distribution
, 1995
"... We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p ..."
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Cited by 28 (0 self)
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We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain fi of small degree, simulation gives the Hausdorff dimension to several decimal places.
Numeration systems on a regular language
 Theory Comput. Syst
"... Generalizations of linear numeration systems in which IN is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study t ..."
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Cited by 23 (9 self)
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Generalizations of linear numeration systems in which IN is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study the translation and the multiplication by constants as well as the orderdependence of the recognizability. 1
On negative bases
"... Abstract. We study expansions in noninteger negative base −β introduced by Ito and Sadahiro [7]. Using countable automata associated with (−β)expansions, we characterize the case where the (−β)shift is a system of finite type. We prove that, if β is a Pisot number, then the (−β)shift is a sofic ..."
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Cited by 16 (1 self)
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Abstract. We study expansions in noninteger negative base −β introduced by Ito and Sadahiro [7]. Using countable automata associated with (−β)expansions, we characterize the case where the (−β)shift is a system of finite type. We prove that, if β is a Pisot number, then the (−β)shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. 1
Number of representations related to a linear recurrent basis
 Acta Arith
, 1999
"... Abstract. We consider a sequence (Gn) satisfying a stationary recurrence relation with a Perron dominant root φ of the corresponding polynomial. We fix q ≥ 2 and study the properties of the summation function Φ(x) = y<x f(y), where f(y) is defined as the number of representations of y in the basi ..."
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Cited by 15 (5 self)
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Abstract. We consider a sequence (Gn) satisfying a stationary recurrence relation with a Perron dominant root φ of the corresponding polynomial. We fix q ≥ 2 and study the properties of the summation function Φ(x) = y<x f(y), where f(y) is defined as the number of representations of y in the basis (Gn) with the digits 0, 1,..., q−1. The main result is the existence of a (unique) periodic function H with period 1 such that Φ(x) = xαH(logφ x) + o(x α), x→∞, where α = logφ q. It is proved that H is positive, continuous and a.e. differentiable. Besides, we reduce the problem of estimating the remainder term and Lipschitz exponent for H to finding a good exponent in the upper bound for f. A special attention is paid to the radix case with Gn = dn. 2 0. General results and notations. We first present a general statement on the number of representations related to a linear recurrent basis. Let r be an integer, r ≥ 1, and let a1, a2,..., ar be reals. We consider a sequence (Gk)k≥0 such that for