We study the palindromic complexity of infinite words $u_{\beta }$, the fixed points of the substitution over a binary alphabet, $\varphi \left(0\right)=0^{a}1$, $\varphi \left(1\right)=0^{b}1$, with $a-1\ge b\ge 1$, which are canonically associated with quadratic non-simple Parry numbers $\beta$.

We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β.

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form $d_{\beta }\left(1\right)=t_1\cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_{\beta }$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_{\beta }$ contains infinitely many palindromes if and only if $t_1=t_2=\cdots =t_{m-1}\ge t_m$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. If $t_m=1$ then $u_{\beta }$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then...

Integer sequences of the form $\lfloor n^c\rfloor$, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference ${\sum }_{n\le x}\phi \left(\lfloor n^c\rfloor \right)-1/c{\sum }_{n\le x^c}\phi \left(n\right)n^{1/c-1}$, which measures the deviation of the mean value of φ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with asymptotic...

For a large class of digital functions $f$, we estimate the sums ${\sum }_{n\le x}\Lambda \left(n\right)f\left(n\right)$ (and ${\sum }_{n\le x}\mu \left(n\right)f\left(n\right)$, where $\Lambda$ denotes the von Mangoldt function (and $\mu$ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

Topological and combinatorial properties of dynamical systems called odometers and arising from number systems are investigated. First, a topological classification is obtained. Then a rooted tree describing the carries in the addition of 1 is introduced and extensively studied. It yields a description of points of discontinuity and a notion of low scale, which is helpful in producing examples of what the dynamics of an odometer can look like. Density of the orbits is also discussed.