Let $f\left(x\right)$ be a power series ${\sum }_{n\ge 1}\zeta \left(n\right)x^{e\left(n\right)}$, where $\left(e\right(n\left)\right)$ is a strictly increasing linear recurrence sequence of non-negative integers, and $\left(\zeta \right(n\left)\right)$ a sequence of roots of unity in ${\overline{\mathbb{Q}}}_p$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ${\mathbb{Q}}_p$ of the elements $f\left({\alpha }_1\right),...,$ $f\left({\alpha }_t\right)$ from ${ℂ}_p$ in terms of the distinct ${\alpha }_1,...,{\alpha }_t\in {\mathbb{Q}}_p$ satisfying $0\<|{\alpha }_{\tau }{|}_p\<1$ for $\tau =1,...,t$. A striking application of our basic result says that, in the case $e\left(n\right)=n$, the set $\left\{f\left(\alpha \right)\right|\alpha \in {\mathbb{Q}}_p,0\<|\alpha {|}_p\<1\}$ is algebraically independent over ${\mathbb{Q}}_p$ if...

For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers ${\gamma }^p$ of algebraic integers $\gamma$ in $K$ exist, such that $\mu \left(\tau \left({\gamma }^p\right)\right)\le X$ for each embedding $\tau$ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu \left(z\right)=max\left(\right|\mathrm{Re}\left(z\right)|,|\mathrm{Im}\left(z\right)\left|\right)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_{p,K}{X}^{n/p}+R_{p,K}\left(X\right)$, with sharp estimates for the remainder $R_{p,K}\left(X\right)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result...

Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have $$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...

Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact...

We give explicit non-recursive formulas to compute the Josephus-numbers $j(n,2,i)$ and $j(n,3,i)$ and explicit upper and lower bounds for $j(n,k,i)$ (where $k\ge 4$) which differ by $2k-2$ (for $k=4$ the bounds are even better). Furthermore we present a new fast algorithm to calculate $j(n,k,i)$ which is based upon the mentioned bounds.

For $a\in {\mathbf{R}}^n\setminus \left\{0\right\}$ and $\Omega$ an open bounded subset of ${\mathbf{R}}^n$ definie $L^p(\Omega ,a)$ as the closed subset of $L^p\left(\Omega \right)$ consisting of all functions that are constant almost everywhere on almost all lines parallel to $a$. For a given set of directions $a^{\nu }\in {\mathbf{R}}^n\setminus \left\{0\right\}$, $\nu =1,...,m$, we study for which $\Omega$ it is true that the vector space $$(*)L^p(\Omega ,a^1)+\cdots +L^p(\Omega ,a^m)\text{is}\text{a}\text{closed}\text{subspace}\text{of}{L}^p\left(\Omega \right).$$ This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...