Hausdorff dimension, Mean quadratic variation of infinite self-similar measures

Hausdorff dimension of the large values of Weyl sums

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2πi (x_1 n+\ldots +x_d n^{d})) \right| \ge N^α $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion …

The infinite derivatives of Okamoto's self-affine functions: an application of beta-expansions

Okamoto's one-parameter family of self-affine functions $F_a: [0,1]\to[0,1]$, where $0<a<1$, includes the continuous nowhere differentiable functions of Perkins ($a=5/6$) and Bourbaki/Katsuura ($a=2/3$), as well as the Cantor function ($a=1/2$). The main purpose of this article is to characterize the set of points at which $F_a$ has an infinite derivative. We compute the Hausdorff dimension of this set for the case $a\leq 1/2$, and estimate it for $a>1/2$. For all $a$, we determine the Hausdorff dimension of the sets of points where: (i) $F_a'=0$; and (ii) $F_a$ has neither a finite nor an inf…

Large Weyl sums and Hausdorff dimension

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $α\in (1/2,1)$ achieve the order at least $N^α$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $d\ge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $α$ is close to $1$. We also obtain a nearly tight bound in a s…

Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Generalising a construction of Falconer, we consider classes of $G_δ$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes. As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. F…

无限生成吸引子的中间维数

Intermediate dimensions of infinitely generated attractors

研究无限生成吸引子的中间维数，证明共形情形下其等于极限集豪斯多夫维数与不动点集中间维数的最大值，并给出投影、分数布朗图像等应用及一般上界。