arXiv subjects

Roope Anttila

Publications and source records attributed to Roope Anttila.

9 recordsLinked to original sources

arXiv2026

One-dimensional Dirichlet forms with prescribed Hölder regularity

We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $δ\in (0,1]$ there exists a metric measure space $(X,d,μ)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(μ)$ with the property that $δ$ is the supremum of $α\in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $α$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $δ= 1$. In our construction, the value $δ$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.

math.FA↗

arXiv2026

Badly approximable points on non-linear carpets

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.

math.NT↗

arXiv2026

Dvoretzky covering problem for general measures

We study the Dvoretzky covering problem for random covering sets driven by general Borel probability measures. As our main result, we solve the problem of covering analytic sets by random covering sets generated by arbitrary Borel probability measures on the real line. Prior to this work, a complete solution was not known for any singular measure. Our solution is potential theoretic and involves a generalisation of a notion of capacity in the work of Kahane, who solved the problem of covering compact sets in the classical setting where the random covering process is driven by the Lebesgue measure on the unit circle. One of our key innovations is a simple but powerful application of the Jankov-von Neumann uniformisation theorem, which we believe to have interest outside of this work. In addition, we determine the critical exponent for the covering problem for polynomially decreasing sequences $(cn^{-t})_n$ for random covering sets driven by Borel probability measures on $\mathbb{R}^d$. At exactly the critical exponent, the covering property generally depends on the constant $c>0$, and as an application of our main result, we determine the critical constant for random covering sets driven by natural measures on strongly separated self-conformal sets on the line. The critical constant depends on the multifractal structure of the average densities of the measure, and the result is new even for the simplest case of the Hausdorff measure on the Cantor set.

math.PR↗

arXiv2025

Quasisymmetric mappings on two variants of fractal percolation

We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

math.MG↗

arXiv2024

Fibre stability for dominated self-affine sets

Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in π_{V^\bot}(E)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Here, $\operatorname{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\max_{x\in π_{V^\bot}(K)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.

math.DS↗

arXiv2024

Level sets of prevalent Hölder functions

We study the level sets of prevalent Hölder functions. For a prevalent $α$-Hölder function on the unit interval, we show that the upper Minkowski dimension of every level set is bounded from above by $1-α$ and Lebesgue positively many level sets have Hausdorff dimension equal to $1-α$.

math.CA↗

arXiv2024

Pointwise Assouad dimension for measures

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension differs from the global counterpart, but in many classical cases, it exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension at almost every point. We also compute the Assouad dimension of invariant measures with place dependent probabilities supported on strongly separated self-conformal sets.

math.CA↗

arXiv2024

Multifractal analysis for the pointwise Assouad dimension of self-similar measures

We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full Hausdorff dimension, despite the fact that they can in general be non-doubling in a set of full Hausdorff measure. More generally, we carry out multifractal analysis by determining the Hausdorff dimension of the level sets of the pointwise Assouad dimension.

math.DS↗

arXiv2023

Slices of the Takagi function

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that the Marstrand's slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the $x$-axis lifted to the graph is at least one.

math.CA↗