Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as $$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$ where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then $${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$ moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows ...

We study the Siegel-Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey-$s$, $s\>0$ category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey-$s$ formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin, for the analytic germ.

We study the solvability of equations associated with a complex vector field $L$ in ${ℝ}^2$ with $C^{\infty }$ or $C^{\omega }$ coefficients. We assume that $L$ is elliptic everywhere except on a simple and closed curve $\Sigma$. We assume that, on $\Sigma$, $L$ is of infinite type and that $L\wedge \overline{L}$ vanishes to a constant order. The equations considered are of the form $Lu=pu+f$, with $f$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $L\wedge \overline{L}$ is $\>1$, the equation $Lu=f$ is solvable in the $C^{\infty }$ category but not in the...

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\right(x),x)]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

The $L^1$ norm of a trigonometric polynomial with $N$ non zero coefficients of absolute value not less than 1 exceeds a fixed positive multiple of $C\left(\log N\right)/(\log \log N{)}^2.$