Related papers: Dimension 1 sequences are close to randoms

In this paper we consider functions of the type $$f(x) = \sum_{n=0}^\infty a_n g(b_nx+\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \geq b >1$, $a^2b> 1$…

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional…

According to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. We extend Wik's result to the multidimensional case.

Let $\theta$ be an irrational number and $\varphi: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\varphi(\theta) =\Big\{y \in \mathbb R: \|n\theta- y\|<\varphi(n), \ {\text{for infinitely…

Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes of order greater than s when r tends to 0. In this paper we…

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can…

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially…

A connected compact subset $E$ of $\mathbb{R}^N$ is said to be a strict Whitney set if there exists a real-valued $C^1$ function $f$ on $\mathbb{R}^N$ with $\nabla f|_E\equiv 0$ such that $f$ is constant on no non-empty relatively open…

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any $0\le s\le 1$, there exists some set on $\mathbb{R}$ with…

Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of…

Metric mean dimension and mean Hausdorff dimension depend on metrics. In this paper, we investigate the continuity of the metric mean dimension and mean Hausdorff dimension concerning the metrics for amenable group actions, which extends…

Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff…

We prove that for any $1 \le k<n$ and $s\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 <…

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has…

Let the sequence $\{t_n\}_{n=1}^{\infty}$ of reals satisfy the condition $ \frac{t_{n+1}}{t_n} \ge 1+ \frac{\gamma}{n^\beta},0\le \beta <1, \gamma >0. $ Then the set $ \{\alpha \in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n…

The decay rate of Riesz capacity as the exponent increases to the dimension of the set is shown to yield Hausdorff measure. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For…

We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of…

Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of…

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley…

Various methods have been proposed to approximate a solution to the truncated Hausdorff moment problem. In this paper, we establish a method of comparison for the performance of the approximations. Three ways of producing random moment…