Thue-Morse at multiples of an integer |
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Authors: | Johannes F Morgenbesser Jeffrey Shallit |
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Institution: | a Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria b Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France c School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada |
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Abstract: | Let t=(tn)n?0 be the classical Thue-Morse sequence defined by , where s2 is the sum of the bits in the binary representation of n. It is well known that for any integer k?1 the frequency of the letter “1” in the subsequence t0,tk,t2k,… is asymptotically 1/2. Here we prove that for any k there is an n?k+4 such that tkn=1. Moreover, we show that n can be chosen to have Hamming weight ?3. This is best in a twofold sense. First, there are infinitely many k such that tkn=1 implies that n has Hamming weight ?3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s2 is replaced by sb for an arbitrary base b?2. |
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Keywords: | 11N25 11A63 68R15 |
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