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 $f\left(x\right)\sim \Sigma a_n{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_n\cos 2\pi nx$, where the $a{*}_n$ are the numbers $\left|a_n\right|$ rearranged so that $a_n^{*}\searrow 0$. Then for any convex increasing $\psi$, $\parallel \psi \left(\right|f{|}^2{\parallel }_1\le \parallel \psi \left(20\right|f*{|}^2{\parallel }_1$. The special case $\psi \left(t\right)=t^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_q\le 5\parallel f*{\parallel }_q$ an equivalent of Littlewood.

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...

Let $x\in ]0,1]$ and $p_n/q_n,n\ge 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients ${\theta }_n={\theta }_n\left(x\right)$ by $q_{n}x-p_n,n\ge 1$. In [BBDK] the limiting distribution of the sequence ${\left({\theta }_n\right)}_{n\ge 1}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n,{\theta }_n$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution...

Let $T:x\mapsto x+g$ be an ergodic translation on the compact group $C$ and $M\subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mathbf{a}:\mathbb{Z}\mapsto \{0,1\}$ defined by $\mathbf{a}\left(k\right)=1$ if $T^k\left(0_C\right)\in M$ and $\mathbf{a}\left(k\right)=0$ otherwise, is called a Hartman sequence. This paper studies the growth rate of $P_{\mathbf{a}}\left(n\right)$, where $P_{\mathbf{a}}\left(n\right)$ denotes the number of binary words of length $n\in \mathbb{N}$ occurring in $\mathbf{a}$. The growth rate is always subexponential and this result is optimal. If $T$ is an ergodic translation $x\mapsto x+\alpha$ $(\alpha =({\alpha }_1,...,{\alpha }_s))$ on ${𝕋}^s$ and $M$ is a box with...