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Alex Rutar

Publications and source records attributed to Alex Rutar.

At least 19 recordsLinked to original sources
arXiv2026

Visibility problem in the plane

We disprove the visibility conjecture in the plane and prove the sharp upper bound for the almost-sure dimension of visible parts. Precisely, let $K \subset \mathbb{R}^{2}$ be a compact set. For $\sigma \in S^{1}$, let $\mathrm{Vis}_{\sigma}(K) \subset K$ be the visible part of $K$ in direction $\sigma$. We prove that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_{\sigma}(K) \leq \tfrac{3}{2}$ for $\mathcal{H}^{1}$ almost every $\sigma \in S^{1}$. This is sharp: we construct a compact set $K \subset \mathbb{R}^{2}$ such that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_{\sigma}(K)\geq \tfrac{3}{2}$ for all $\sigma \in S^1$.

math.CO↗
arXiv2026

Attainable forms of lower spectra

Let $d\in\mathbb{N}$ and $\varphi\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^{\theta}_{\mathrm{L}} F$ satisfies $(1-\theta)\operatorname{dim}^{\theta}_{\mathrm{L}} F = \varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for all $\lambda,\theta\in(0,1)$, \begin{equation*} \varphi(\theta) \leq \varphi(\lambda\theta) - \theta \varphi(\lambda) \leq (1-\theta) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^{\theta}_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgov\'anyi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.

math.CA↗
arXiv2025

Two-scale branching functions and inhomogeneous attractors

We introduce the notion of a two-scale branching function associated with an arbitrary metric space, which encodes the lower and upper box dimensions as well as the Assouad spectrum. If the metric space is quasi-doubling, this function is approximately Lipschitz. We fully classify the attainable Lipschitz two-scale branching functions, which gives a new proof of the classification of Assouad spectra due to the second author. We then study inhomogeneous self-conformal sets satisfying standard separation conditions. We show that the two-scale branching function of the attractor is given explicitly in terms of the two-scale branching function of the condensation set and the Hausdorff dimension of the homogeneous attractor. In particular, this gives formulas for the lower box dimension and the Assouad spectrum of the attractor.

math.DS↗
arXiv2025

A fractal local smoothing problem for the wave equation

For any given set $E\subset [1,2]$, we discuss a fractal frequency-localized version of the $L^p$ local smoothing estimates for the half-wave propagator with times in $E$. A conjecture is formulated in terms of a quantity involving the Assouad spectrum of $E$ and the Legendre transform. We validate the conjecture for radial functions. We also prove a similar result for fractal-time $L^2\to L^q$ and square function bounds, for arbitrary $L^2$ functions and general time sets. We formulate a conjecture for $L^p\to L^q$ generalizations.

math.CA↗
arXiv2024

On the uniformity and size of microsets

We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set $K\subset\mathbb{R}^d$ with Assouad dimension arbitrarily close to $d$ such that every microset of $K$ has no Ahlfors--David regular subset with dimension strictly larger than $0$. This answers a question of Orponen. Then, we show that for any non-empty compact set $K\subset\mathbb{R}^d$ with lower dimension $\beta$, there is a microset $E$ of $K$ with finite $\beta$-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--K\"aenm\"aki--Yu, who previously obtained a similar result concerning the upper box dimension.

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 $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in \pi_{V^\bot}(E)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ 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 $$\max_{x\in \pi_{V^\bot}(K)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.

math.DS↗
arXiv2024

Tangents and slices of self-affine carpets

We study the fine scaling properties of planar self-affine carpets. For Gatzouras--Lalley carpets, we give a precise formula for maximal Hausdorff dimension of a tangent in terms of the Hausdorff dimension of the projection and the Assouad dimension of the corresponding vertical slice. Using regularity properties for the Assouad dimension of non-autonomous self-similar sets, this implies that the set of points with tangents that are as large as possible has full Hausdorff measure, at the critical exponent. On the other hand, we give an explicit example of a Bara\'nski carpet for which the Hausdorff dimension of the set of points for which there exists a maximal tangent has Hausdorff dimension strictly less than the Hausdorff dimension of the original carpet.

math.DS↗
arXiv2024

Regularity of non-autonomous self-similar sets

Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this note is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in $\mathbb{R}^d$ satisfying a certain ``bounded neighbourhood'' condition, which generalizes earlier work of Li--Li--Miao--Xi and Olson--Robinson--Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.

math.DS↗
arXiv2024

Lower box dimension of infinitely generated self-conformal sets

Let $\Lambda$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $\Lambda$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq \max \{\dim_{\mathrm H} \Lambda, \underline{\dim}_{\mathrm B} F\}. \] In particular, this provides the first examples of sets of continued fraction expansions with restricted digits for which the box dimension does not exist. More generally, we establish an explicit asymptotic formula for the covering numbers $N_r(\Lambda)$ in terms of $\dim_{\mathrm H}\Lambda$ and the covering function $r\mapsto N_r(F)$, where $N_r(\cdot)$ denotes the least number of open balls of radius $r$ required to cover a given set. Such finer scaling information is necessary: in general, the lower box dimension $\underline{\dim}_{\mathrm B} \Lambda$ is not a function of the Hausdorff dimension of $\Lambda$ and the upper and lower box dimensions of $F$, and we prove sharp bounds for $\underline{\dim}_{\mathrm B} \Lambda$ in terms of these three quantities.

math.DS↗
arXiv2024

Assouad spectrum of Gatzouras-Lalley carpets

We study the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets. More precisely, we establish a formula for the Assouad spectrum of all Gatzouras-Lalley carpets as the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Our formula implies a number of novel properties for the Assouad spectrum not previously observed for dynamically invariant sets; in particular, the Assouad spectrum can be a non-trivial differentiable function on the entire domain $(0,1)$ and can be strictly concave on open intervals. Our proof introduces a general framework for covering arguments using techniques developed in the context of multifractal analysis, including the method of types from large deviations theory and Lagrange duality from optimisation theory.

math.DS↗
arXiv2023

Multifractal analysis via Lagrange duality

We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode the parametric geometry of the Lagrange dual associated with the constrained variational principle. We also give a direct derivation of the Hausdorff dimension of the level sets of the upper and lower local dimensions by exploiting certain weak uniformity properties of the space of Bernoulli measures.

math.DS↗
arXiv2023

Tangents of invariant sets

We study the fine scaling properties of sets satisfying various weak forms of invariance. For general attractors of possibly overlapping bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the Hausdorff dimension of a tangent at some point in the attractor. Under the additional assumption of self-conformality, we moreover prove that this property holds for a subset of full Hausdorff dimension.

math.DS↗
arXiv2023

Interpolating with generalized Assouad dimensions

The $\phi$-Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key properties of the $\phi$-Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space $F$ and $\alpha\in\mathbb{R}$ satisfying $\overline{\operatorname{dim}}_{\mathrm{B}}F<\alpha\leq\operatorname{dim}_{\mathrm{A}} F$ that there is a function $\phi$ so that the $\phi$-Assouad dimension of $F$ is equal to $\alpha$. We further show that the "upper" variant of the dimension is fully determined by the $\phi$-Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the $\phi$-Assouad dimensions for Galton--Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. This result follows from two results which may be of general interest: a sharp large deviations theorem for Galton--Watson processes with bounded offspring distribution, and a Borel--Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the $\phi$-Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.

math.CA↗
arXiv2023

Multifractal analysis of measures arising from random substitutions

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the $L^q$-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the $L^q$-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word $L^q$-spectrum of a random substitution and show that this coincides with the $L^q$-spectrum of the corresponding frequency measure for all $q \geq 0$. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.

math.DS↗
arXiv2022

Assouad-type Dimensions of Overlapping Self-affine Sets

We study the Assouad and quasi-Assoaud dimensions of dominated rectangular self-affine sets in the plane. In contrast to previous work on the dimension theory of self-affine sets, we assume that the sets satisfy certain separation conditions on the projection to the principal axis, but otherwise have arbitrary overlaps in the plane. We introduce and study regularity properties of a certain symbolic non-autonomous iterated function system corresponding to "symbolic slices" of the self-affine set. We then establish dimensional formulas for the self-affine sets in terms of the dimension of the projection along with the maximal dimension of slices orthogonal to the projection. Our results are new even in the case when the self-affine set satisfies the strong separation condition: in fact, as an application, we show that self-affine sets satisfying the strong separation condition can have distinct Assouad and quasi-Assouad dimensions, answering a question of the first named author.

math.DS↗
arXiv2022

Attainable forms of Assouad spectra

Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^\theta F=\varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for every $0<\lambda<\theta<1$, \[0\leq (1-\lambda)\varphi(\lambda)-(1-\theta)\varphi(\theta)\leq (\theta-\lambda)\varphi\Bigl(\frac{\lambda}{\theta}\Bigr).\] In particular, the following behaviours which have not previously been witnessed in any examples are possible: the Assouad spectrum can be non-monotonic on every open set, and can fail to be H\"older in a neighbourhood of 1.

math.CA↗
arXiv2021

Attainable forms of intermediate dimensions

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function $h(\theta)$ to be realized as the intermediate dimensions of a bounded subset of $\mathbb{R}^d$. This condition is a straightforward constraint on the Dini derivatives of $h(\theta)$, which we prove is sharp using a homogeneous Moran set construction.

math.MG↗