807 research outputs found

### Infinite products involving binary digit sums

Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. 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 $n$
and the sequence $u_n$ 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 $f$. We also obtain some new identities
similar to the Woods-Robbins product.Comment: Accepted in Proc. AMMCS 2017, updated according to the referees'
comment

### 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

### Summation of Series Defined by Counting Blocks of Digits

We discuss the summation of certain series defined by counting blocks of
digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes
the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1}
s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2$. 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

### Supercritical holes for the doubling map

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

### On univoque Pisot numbers

We study Pisot numbers $\beta \in (1, 2)$ which are univoque, i.e., such that
there exists only one representation of 1 as $1 = \sum_{n \geq 1}
s_n\beta^{-n}$, with $s_n \in \{0, 1\}$. 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

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