We characterize the power series ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ with the geometric property that, for sufficiently many points $z$, $\left|z\right|=1$, a circle $C\left(z\right)$ contains infinitely many partial sums. We show that ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ is a rational function of special type; more precisely, there are $t\in ℝ$ and $n_0$, such that, the sequence $c_n{e}^{\mathrm{int}}$, $n\ge n_0$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. We are led to consider special families of...

Given a large prime number $p$ and a rational function $r\left(X\right)$ defined over ${𝔽}_p=\mathbb{Z}/p\mathbb{Z}$, we investigate the size of the set $\left\{x\in {𝔽}_p:\tilde{r}\left(x\right)\>\tilde{r}(x+1)\right.$, where $\tilde{r}\left(x\right)$ and $\tilde{r}(x+1)$ denote the least positive representatives of $r\left(x\right)$ and $r(x+1)$ in $\mathbb{Z}$ modulo $p\mathbb{Z}$.

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 $$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

Let $X\to B$ an elliptic fibration with general fibre $F$. Let $n_e,n_s,n_a,n_v$ be the minima of the non-zero intersection numbers $(ℒ,F)$ where $ℒ$ runs successively through the following sets: effective divisors on $X$, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X\to B$. We prove that $n_e=n_s$ if and only if $n_e\ge 2m$ and that $n_a=n_v$ if and only if $n_a\ge 3m$.

Let $\Gamma$ be a non-pluripolar set in ${ℂ}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma$. Let $\left\{P_n\right\}$ be a sequence of polynomials with $deg{P}_n\le d_n(d_n\<d_{n+1})$ such that $$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}\<1,z\in \Gamma .$$ We show that if $$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$ where $E$ is a set in ${ℂ}^N$ such that the global extremal function $V_E\equiv 0$ in ${ℂ}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and $$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}\<1$$ for every compact set $K\subset G_f$. If, moreover, the sequence $\{d_{n+1}/d_n\}$ is bounded then $G_f={ℂ}^N$. If $E$ is a closed...

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...