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A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ( x ) + 1 / ( d ( x ) d ( x ) ) + . . . + 1 / ( d ( x ) d ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote the Hausdorff...

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form x = i = 1 c i q - i with integer coefficients c i satisfying 0 c i < q , i ≥ 1. In this case we say that ( c i ) = c c . . . is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...

Applications de la notion d'entropie au développement d'un nombre réel dans une base de Pisot

Anne Bertrand-Mathis (1985)

Annales de l'institut Fourier

Soit θ un nombre de Pisot de degré s  ; nous avons montré précédemment que l’endomorphisme du tore T s dont θ est valeur propre est facteur du θ -shift bilatéral par une application continue q s  ; nous prouvons ici (théorème 1) que l’application q s conserve l’entropie de toute mesure invariante sur le θ -shift. Ceci permet de définir l’entropie d’un nombre dans la base θ et d’en étudier la stabilité. Nous généralisons également des résultats de Kamae, Rauzy et Bernay.

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 &gt; α , ...

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