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Aleksi Pyörälä

Publications and source records attributed to Aleksi Pyörälä.

10 recordsLinked to original sources
arXiv2026

Visible parts and projections of the affine fractal percolation

We study the visible parts and orthogonal projections of the affine fractal percolation. We show that for the affine fractal percolation $E$, conditioned on $E \neq \varnothing$, almost surely the dimension of the visible part is equal to $\min\{1, \dim_{\rm H} E\}$ from all directions except for the major contracting direction. The visible part from the major contracting direction is notably different from the other directions, and we provide a formula for its Hausdorff and box dimensions. In addition, we show that almost surely conditioned on $E \neq \varnothing$, $\dim_{\rm H} π(E) =\min\{ 1, \dim_{\rm H} E\}$ for every orthogonal projection $π$ excluding the one onto the major contracting direction, and given $\dim_{\rm H} E >1$, these projections contain intervals.

math.CA↗
arXiv2026

On dissonance and orthogonal projections of self-conformal measures

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. We establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. In addition, we show that $\dim μ\circπ^{-1} = \min\{ k, \dim μ\}$ for every ortohogonal projection $π:\mathbb{R}^d\to\mathbb{R}^k$, $0<k<d$, when either $d=2$ and $μ$ is not self-similar and not supported on a line, or $d\geq 3$ and $μ$ is totally non-linear and not supported on a smooth hypersurface.

math.DS↗
arXiv2026

Resonance between planar self-affine measures

We show that if $\lbrace φ_i\rbrace_{i\in Γ}$ and $\lbrace ψ_j\rbrace_{j\inΛ}$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $μ$ and $ν$, the inequality $$\dim_{\rm H}(μ*ν) < \min \lbrace 2, \dim_{\rm H} μ+ \dim_{\rm H} ν\rbrace$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of $φ_i$ and $ψ_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.

math.CA↗
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↗
arXiv2025

On Fourier transforms of fractal measures on the parabola

Let $s \in [0,1]$ and $t \in [0,\min\{3s,s + 1\})$. Let $σ$ be a Borel measure supported on the parabola $\mathbb{P} = \{(x,x^{2}) : x \in [-1,1]\}$ satisfying the $s$-dimensional Frostman condition $σ(B(x,r)) \leq r^{s}$. Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \geq 1$ such that $$\|\hatσ\|_{L^{p}(B(R))} \leq C_{s,t}R^{(2 - t)/p}, \qquad R \geq 1.$$ Moreover, when $s \geq 2/3$ and $t \in [0,s + 1)$, the previous inequality is true for $p \geq 6$. We also obtain the following fractal geometric counterpart of the previous results. If $K \subset \mathbb{P}$ is a Borel set with $\dim_{\mathrm{H}} K = s \in [0,1]$, and $n \geq 1$ is an integer, then $$ \dim_{\mathrm{H}}(nK) \geq \min\{3s - s \cdot 2^{-(n - 2)},s + 1\}.$$

math.CA↗
arXiv2025

The dimension of projections of planar diagonal self-affine measures

We show that if $μ$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $$ \dim_{\rm H} μ\circ π^{-1}= \min\lbrace 1,\dim_{\rm H}μ\rbrace $$ for any non-principal orthogonal projection $π$.

math.DS↗
arXiv2023

Scaling limits of self-conformal measures

We show that any self-conformal measure $μ$ on $\mathbb{R}$ is uniformly scaling and generates an ergodic fractal distribution. This generalizes existing results by removing the need for any separation condition. We also obtain applications to the prevalence of normal numbers in self-conformal sets, the resonance between self-conformal measures on the line, and projections of self-affine measures on carpets.

math.DS↗
arXiv2020

Covering the Sierpiński carpet with tubes

We show that non-trivial $\times N$-invariant sets in $[0,1]^d$, such as the Sierpiński carpet and the Sierpiński sponge, are tube-null, that is, they can be covered by a union of tubular neighbourhoods of lines of arbitrarily small total volume. This introduces a new class of tube-null sets of dimension strictly between $d-1$ and $d$. We utilize ergodic-theoretic methods to decompose the set into finitely many parts, each of which projects onto a set of Hausdorff dimension less than $1$ in some direction. We also discuss coverings by tubes for other self-similar sets, and present various applications.

math.CA↗