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Automata, algebraicity and distribution of sequences of powers

Jean-Paul Allouche, Jean-Marc Deshouillers, Teturo Kamae, Tadahiro Koyanagi (2001)

Annales de l’institut Fourier

Let K be a finite field of characteristic p . Let K ( ( x ) ) be the field of formal Laurent series f ( x ) in x with coefficients in K . That is, f ( x ) = n = n 0 f n x n with n 0 𝐙 and f n K ( n = n 0 , n 0 + 1 , ) . We discuss the distribution of ( { f m } ) m = 0 , 1 , 2 , for f K ( ( x ) ) , where { f } : = n = 0 f n x n K [ [ x ] ] denotes the nonnegative part of f K ( ( x ) ) . This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic distribution for f with f n 0 for some n < 0 . This distribution is...

Automates finis et ensembles normaux

Christian Mauduit (1986)

Annales de l'institut Fourier

Soit u = ( u n ) n N une suite strictement croissante d’entiers reconnaissable par un automate fini. Nous montrons qu’une condition nécessaire et suffisante pour que l’ensemble normal associé a u soit exactement R Q est que l’un au moins des sommets qui reconnaît la suite u soit précédé dans le graphe de l’automate par un sommet possédant au moins deux circuits fermés distincts. Cette condition peut se traduire quantitativement en disant que la suite u doit être plus “dense” que toute suite exponentielle.

Basic properties of shift radix systems.

Akiyama, Shigeki, Borbély, Tibor, Brunotte, Horst, Pethő, Attila, Thuswaldner, Jörg M. (2006)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Besicovitch subsets of self-similar sets

Ji-Hua Ma, Zhi-Ying Wen, Jun Wu (2002)

Annales de l’institut Fourier

Let E be a self-similar set with similarities ratio r j ( 0 j m - 1 ) and Hausdorff dimension s , let p ( p 0 , p 1 ) ... p m - 1 be a probability vector. The Besicovitch-type subset of E is defined as E ( p ) = x E : lim n 1 n k = 1 n χ j ( x k ) = p j , 0 j m - 1 , where χ j is the indicator function of the set { j } . Let α = dim H ( E ( p ) ) = dim P ( E ( p ) ) = j = 0 m - 1 p j log p j j = 0 m - 1 p i log r j and g be a gauge function, then we prove in this paper:(i) If p = ( r 0 s , r 1 s , , r m - 1 s ) , then s ( E ( p ) ) = s ( E ) , 𝒫 s ( E ( p ) ) = 𝒫 s ( E ) , moreover both of s ( E ) and 𝒫 s ( E ) are finite positive;(ii) If p is a positive probability vector other than ( r 0 s , r 1 s , , r m - 1 s ) , then the gauge functions can be partitioned as follows g ( E ( p ) ) = + lim ¯ t 0 log g ( t ) log t α ; g ( E ( p ) ) = 0 lim ¯ t 0 log g ( t ) log t > α , ...

Beta expansion of Salem numbers approaching Pisot numbers with the finiteness property

Hachem Hichri (2015)

Acta Arithmetica

It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.

Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

Bill Mance, Jimmy Tseng (2013)

Acta Arithmetica

We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition,...

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