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Complexity of Hartman sequences

Christian SteinederReinhard Winkler — 2005

Journal de Théorie des Nombres de Bordeaux

Let T : x x + g be an ergodic translation on the compact group C and M C a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence a : { 0 , 1 } defined by a ( k ) = 1 if T k ( 0 C ) M and a ( k ) = 0 otherwise, is called a Hartman sequence. This paper studies the growth rate of P a ( n ) , where P a ( n ) denotes the number of binary words of length n occurring in a . The growth rate is always subexponential and this result is optimal. If T is an ergodic translation x x + α ( α = ( α 1 , ... , α s ) ) on 𝕋 s and M is a box with side lengths...

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