We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi < 0$, we obtain a family of conditional measures $\nu_{\bf y}$ on the line, parametrized by ${\bf y} = (y_1,y_2,\dotsc)$, the sequence of errors. Our main result is that if $h > |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h < |\chi|$, then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies $K, T_1,\ldots, T_s$. (a) Let $\varepsilon\,{\in}\, (0,1)$ and let $x_1,\ldots, x_N$ be chosen from K. Is it true that if $N\,{\geq}\, C(\varepsilon )n\log n$, then \[\left\| I-\frac{1}{NL_K^2}\sum_{i=1}^Nx_i\otimes x_i\right\|<\varepsilon\] with probability greater than $1\,{-}\,\varepsilon $? (b) Let $x_i$ be chosen from $T_i$. Is it true that the unconditional norm \[\|{\bf t}\|=\int_{T_1}\!{\ldots}\int_{T_s}\left\|\sum_{i=1}^st_ix_i\right\|_K\,dx_s\ldots dx_1\] is well comparable to the Euclidean norm in ${\mathbb R}^s$? (c) Let $x_1,\ldots, x_N$ be chosen from K. Let ${\mathbb E}\,(K,N):={\mathbb E}\,|{\rm conv}\{ x_1,\ldots, x_N\}|^{1/n}$ be the expected volume radius of their convex hull. Is it true that ${\mathbb E}\,(K,N)\,{\simeq}\, {\mathbb E}\,(B(n),N)$ for all N, where $B(n)$ is the Euclidean ball of volume 1?

It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.