For a positive integer $n$ and a real number $\xi$, let ${\lambda }_n\left(\xi \right)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $\left|\right|q\xi \left|\right|,\left|\right|q{\xi }^2\left|\right|,...,\left|\right|q{\xi }^n\left|\right|$ are all less than $q^{-\lambda }$. Here, $\left|\right|·\left|\right|$ denotes the distance to the nearest integer. We study the set of values taken by the function ${\lambda }_n$ and, more generally, we are concerned with the joint spectrum of $({\lambda }_1,...,{\lambda }_n,...)$. We further address several open problems.

Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes...

Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of ${\mathbb{P}}_1$ of degree $2$, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\mathrm{PGL}}_2\left(R_S\right)$, admitting a periodic point in ${\mathbb{P}}_1\left(K\right)$ of order $\>3$. Also, all but finitely many classes with a periodic point in ${\mathbb{P}}_1\left(K\right)$ of order $3$ are parametrized by an irreducible...

For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_{k-1}+F_{k-2}+\cdots +F_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m\left(n\right)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m\left(n\right)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the...

We prove that, for a sequence of positive numbers δ(n), if $n^{1/2}\delta \left(n\right)\neg \to \infty$ as $n\to \infty$, to guarantee that the modified Szász-Mirakjan operators $S_{n,\delta }(f,x)$ converge to f(x) at every point, f must be identically zero.