1,129 research outputs found

    Infinite products involving binary digit sums

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    Let (un)n0(u_n)_{n\ge 0} denote the Thue-Morse sequence with values ±1\pm 1. The Woods-Robbins identity below and several of its generalisations are well-known in the literature \begin{equation*}\label{WR}\prod_{n=0}^\infty\left(\frac{2n+1}{2n+2}\right)^{u_n}=\frac{1}{\sqrt 2}.\end{equation*} No other such product involving a rational function in nn and the sequence unu_n seems to be known in closed form. To understand these products in detail we study the function \begin{equation*}f(b,c)=\prod_{n=1}^\infty\left(\frac{n+b}{n+c}\right)^{u_n}.\end{equation*} We prove some analytical properties of ff. We also obtain some new identities similar to the Woods-Robbins product.Comment: Accepted in Proc. AMMCS 2017, updated according to the referees' comment

    Summation of Series Defined by Counting Blocks of Digits

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    We discuss the summation of certain series defined by counting blocks of digits in the BB-ary expansion of an integer. For example, if s2(n)s_2(n) denotes the sum of the base-2 digits of nn, we show that n1s2(n)/(2n(2n+1))=(γ+log4π)/2\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J. Number Theor

    Enumeration and Decidable Properties of Automatic Sequences

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    We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

    Supercritical holes for the doubling map

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    For a map S:XXS:X\to X and an open connected set (== a hole) HXH\subset X we define JH(S)\mathcal J_H(S) to be the set of points in XX whose SS-orbit avoids HH. We say that a hole H0H_0 is supercritical if (i) for any hole HH such that H0ˉH\bar{H_0}\subset H the set JH(S)\mathcal J_H(S) is either empty or contains only fixed points of SS; (ii) for any hole HH such that \barH\subset H_0 the Hausdorff dimension of JH(S)\mathcal J_H(S) is positive. The purpose of this note to completely characterize all supercritical holes for the doubling map Tx=2xmod1Tx=2x\bmod1.Comment: This is a new version, where a full characterization of supercritical holes for the doubling map is obtaine

    On univoque Pisot numbers

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    We study Pisot numbers β(1,2)\beta \in (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1=n1snβn1 = \sum_{n \geq 1} s_n\beta^{-n}, with sn{0,1}s_n \in \{0, 1\}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.Comment: Accepted by Mathematics of COmputatio

    The Critical Exponent is Computable for Automatic Sequences

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    The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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