Newman's phenomenon for generalized Thue-Morse sequences |
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Authors: | M Drmota |
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Institution: | Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria |
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Abstract: | Let tj=(-1)s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721] says that t0+t3+t6+?+t3k>0 for all k?0.In the first part of the paper we show that t1+t4+t7+?+t3k+1<0 and t2+t5+t8+?+t3k+2?0 for k?0, where equality is characterized by means of an automaton. This sharpens results given by Dumont Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145-148]. In the second part we study more general settings. For a,g?2 let ωa=exp(2πi/a) and , where sg(j) denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever a|(g-1). Furthermore, we show that the case a=2 inherits many Newman-like phenomena for every even g?2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Ska?ba Rarified sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609-642] to the general g-case. |
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Keywords: | primary 11B85 secondary 11A63 |
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