1,128 research outputs found
Infinite products involving binary digit sums
Let denote the Thue-Morse sequence with values . 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
and the sequence 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 . 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
We discuss the summation of certain series defined by counting blocks of
digits in the -ary expansion of an integer. For example, if denotes
the sum of the base-2 digits of , we show that . 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
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
For a map and an open connected set ( a hole) we
define to be the set of points in whose -orbit avoids
. We say that a hole is supercritical if (i) for any hole such
that the set is either empty or contains
only fixed points of ; (ii) for any hole such that \barH\subset H_0
the Hausdorff dimension of is positive.
The purpose of this note to completely characterize all supercritical holes
for the doubling map .Comment: This is a new version, where a full characterization of supercritical
holes for the doubling map is obtaine
On univoque Pisot numbers
We study Pisot numbers which are univoque, i.e., such that
there exists only one representation of 1 as , with . 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
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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