arXiv subjects

Richárd Balka

Publications and source records attributed to Richárd Balka.

At least 19 recordsLinked to original sources

arXiv2026

New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions

We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions. We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension. We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar.

math.CA↗

arXiv2025

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 $β$, there is a microset $E$ of $K$ with finite $β$-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--Käenmäki--Yu, who previously obtained a similar result concerning the upper box dimension.

math.MG↗

arXiv2025

Effective eigenvalue approximation from moments for self-adjoint trace-class operators

Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $Λ_n\subset \mathbb{R}$, and we show that it converges to the spectrum of $O$ in the Hausdorff metric under mild conditions. Our set $Λ_n$ only depends on the first $n$ moments of $O$. We show that it can be effectively calculated for physically relevant operators, and it approximates the spectrum well without diagonalization. We prove that using the above method we can converge to the minimal and maximal eigenvalues with super-exponential speed. We also construct monotone increasing lower bounds $q_n$ for the minimal eigenvalue (or decreasing upper bounds for the maximal eigenvalue). This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that $q_n$ can be effectively calculated for a large class of physically relevant operators. This rigorous lower bound $q_n$ tends to the minimal eigenvalue with super-exponential speed provided that $O$ is not positive semidefinite. As a by-product, we obtain computable upper bounds for the $1$-norm of $O$, too. Numerical examples demonstrate the relevance of our approximation in estimating entropy and negativity, which is useful, among others, in quantum optical and in open quantum system models. The results can be directly applicable to problems in quantum information, statistical mechanics, and quantum thermodynamics, where using traditional techniques based on diagonalization is impractical.

quant-ph↗

arXiv2025

Signed null sequences and Hausdorff dimension

We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\notin \ell^1$ in one dimension, then for every $L\in\mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent Lévy vectors.

math.CA↗

arXiv2024

Positivity and entanglement of polynomial Gaussian integral operators

Positivity preservation is an important issue in the dynamics of open quantum systems: positivity violations always mark the border of validity of the model. We investigate the positivity of self-adjoint polynomial Gaussian integral operators $\widehatκ_{PG}$, that is, the multivariable kernel $κ_{PG}$ is a product of a polynomial $P$ and a Gaussian kernel $κ_G$. These operators frequently appear in open quantum systems. We show that $\widehatκ_{PG}$ can be only positive if the Gaussian part is positive, which yields a strong and quite easy test for positivity. This has an important corollary for the bipartite entanglement of the density operators $\widehatκ_{PG}$: if the Gaussian density operator $\widehatκ_G$ fails the Peres-Horodecki criterion, then the corresponding polynomial Gaussian density operators $\widehatκ_{PG}$ also fail the criterion for all $P$, hence they are all entangled. We prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite. We introduce a new preorder $\preceq$ on Gaussian kernels such that if $κ_{G_0}\preceq κ_{G_1}$ then $\widehatκ_{PG_0}\geq 0$ implies $\widehatκ_{PG_1}\geq 0$ for all polynomials $P$. Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of another polynomial Gaussian operators having the same polynomial factor, which might improve any given positivity test by carrying it out on a much larger set of operators. We will show an example that this really can make positivity tests much more sensitive and efficient. This preorder has implication for the entanglement problem, too.

quant-ph↗

arXiv2024

Lipschitz images and dimensions

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if $A$ and $B$ are compact metric spaces and the Hausdorff dimension of $A$ is bigger than the upper box dimension of $B$, then there exist a compact set $A'\subset A$ and a Lipschitz onto map $f\colon A'\to B$. As a corollary we prove that any `natural' dimension in $\mathbb{R}^n$ must be between the Hausdorff and upper box dimensions. We show that if $A$ and $B$ are self-similar sets with the strong separation condition with equal Hausdorff dimension and $A$ is homogeneous, then $A$ can be mapped onto $B$ by a Lipschitz map if and only if $A$ and $B$ are bilipschitz equivalent. For given $α>0$ we also give a characterization of those compact metric spaces that can be obtained as an $α$-Hölder image of a compact subset of $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension.

math.CA↗

arXiv2023

Compact sets with large projections and nowhere dense sumset

We answer a question of Banakh, Jabłońska and Jabłoński by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a $K$ in the unit cube with full projections, that is, such that the projections of $K$ agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for $\ell$-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

math.CA↗

arXiv2021

The range of dimensions of microsets

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $λ_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(λ_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and moreover, $E \cap (0, 1)^d \neq \emptyset$. The main result of the paper is that for a non-empty set $A\subset [0,d]$ there is a compact set $K\subset \mathbb{R}^d$ such that the set of Hausdorff dimensions attained by the microsets of $K$ equals $A$ if and only if $A$ is analytic and contains its infimum and supremum. This answers a question of Fraser, Howroyd, Käenmäki, and Yu. We show that for every compact set $K\subset \mathbb{R}^d$ and non-empty analytic set $A\subset [0,\dim_H K]$ there is a set $\mathcal{C}$ of compact subsets of $K$ which is compact in the Hausdorff metric and $\{\dim_H C: C\in \mathcal{C} \}=A$. The proof relies on the technique of stochastic co-dimension applied for a suitable coupling of fractal percolations with generation dependent retention probabilities. We also examine the analogous problems for packing and box dimensions.

math.CA↗

arXiv2021

Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of $\text{Homeo}([0,1])$ have length $2$. We answer this question in the affirmative. We call $f \in \text{Homeo}([0,1]^d)$ singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of $f \in \text{Homeo}([0,1])$ has length $2$ iff $f$ is singular iff $f$ is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem. We show that for $d \ge 2$ the graph of the generic element of $\text{Homeo}([0,1]^d)$ has infinite $d$-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of $\text{Homeo}([0,1]^d)$ with infinite $d$-dimensional Hausdorff measure is not Haar null. We show that for $d \ge 2$ the generic element of $\text{Homeo}([0,1]^d)$ is singular but not strongly singular. We also show that for $d \ge 2$ almost every element of $\text{Homeo}([0,1]^d)$ is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null). Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.

math.CA↗

arXiv2021

Stability and measurability of the modified lower dimension

The lower dimension $\dim_L$ is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension $\dim_{ML}$ by making the lower dimension monotonic with the simple formula $\dim_{ML} X=\sup\{\dim_L E: E\subset X\}$. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space $X$ let $\mathcal{K}(X)$ denote the metric space of the non-empty compact subsets of $X$ endowed with the Hausdorff metric. As an application of our characterization, we show that the map $\dim_{ML} \colon \mathcal{K}(X)\to [0,\infty]$ is Borel measurable. More precisely, it is of Baire class $2$, but in general not of Baire class $1$. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of $\ell^1$ endowed with the Effros Borel structure.

math.CA↗

arXiv2019

Dimension and measure for generic continuous images

We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key question is `what is the generic dimension of $f(X)$?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, $n$, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of $n$ and the topological dimension of $X$. We also study the typical Hausdorff and packing measures of $f(X)$ and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of $f(X)$ and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.

math.CA↗

arXiv2017

Uniform dimension results for fractional Brownian motion

Kaufman's dimension doubling theorem states that for a planar Brownian motion $\{\mathbf{B}(t): t\in [0,1]\}$ we have $$\mathbb{P}(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1,$$ where $\dim$ may denote both Hausdorff dimension $\dim_H$ and packing dimension $\dim_P$. The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let $0<α<1$ and let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. For a deterministic set $D\subset [0,1]$ consider the following statements: $$(A) \quad \mathbb{P}(\dim_H B(A)=(1/α) \dim_H A \textrm{ for all } A\subset D)=1,$$ $$(B) \quad \mathbb{P}(\dim_P B(A)=(1/α) \dim_P A \textrm{ for all } A\subset D)=1, $$ $$(C) \quad \mathbb{P}(\dim_P B(A)\geq (1/α) \dim_H A \textrm{ for all } A\subset D)=1.$$ We introduce a new concept of dimension, the modified Assouad dimension, denoted by $\dim_{MA}$. We prove that $\dim_{MA} D\leq α$ implies (A), which enables us to reprove a restriction theorem of Angel, Balka, Máthé, and Peres. We show that if $D$ is self-similar then (A) is equivalent to $\dim_{MA} D\leq α$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds iff $\dim_{MA} D\leq α$ or $\dim_H D=0$. The characterization of (A) remains open in general. We prove that $\dim_{MA} D\leq α$ implies (B) and they are equivalent provided that $D$ is analytic. We show that (C) is equivalent to $\dim_H D\leq α$. This implies that if $\dim_H D\leq α$ and $Γ_D=\{E\subset B(D): \dim_H E=\dim_P E\}$, then $$\mathbb{P}(\dim_H (B^{-1}(E)\cap D)=α\dim_H E \textrm{ for all } E\in Γ_D)=1.$$ In particular, all level sets of $B|_{D}$ have Hausdorff dimension zero almost surely.

math.PR↗

arXiv2017

Baum-Katz type theorems with exact threshold

Let $\{X_n\}_{n\geq 1}$ be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centered sequence with a suitable level of negative dependence. We prove Baum-Katz type theorems by only assuming that the variables $X_n$ satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centered, independent random variables. This leads to Marcinkiewicz-Zygmund type strong laws of large numbers with estimate for the rate of convergence.

math.PR↗

arXiv2017

Packing dimension of images and graphs of Gaussian random fields with drift

Let $X=\{(X_1(t),\dots,X_d(t)): t\in \mathbb{R}^n\}$ be a Gaussian random field in $\mathbb{R}^d$ such that $X_1,\dots,X_d$ are independent, centered Gaussian random fields with continuous sample paths. Let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map and let $A\subset \mathbb{R}^n$ be an analytic set. The main goal of the paper is to determine the almost sure value of the packing dimension of the image and graph of $X+f$ restricted to $A$ under a very mild assumption. This generalizes a result of Du, Miao, Wu and Xiao, who calculated the packing dimension of $X(A)$ if $X_1,\dots,X_d$ are independent copies of the same Gaussian random field $X_0$. Provided that $X$ is a fractional Brownian motion, our result is new even if $n=d=1$ and $f$ is continuous, and even if $f\equiv 0$ in the case of graphs. For a fractional Brownian motion $X$ we also obtain the sharp lower bound for the packing dimension of the graph of $X$ over $A$ in terms of the Hurst index of $X$ and the packing dimension of $A$. The analogous result for images was obtained by Talagrand and Xiao.

math.PR↗

arXiv2017

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$. Later on, the author extended the theory for maps from $K$ to $\mathbb{R}^n$. The main goal of this paper is to generalize the relevant results for topological and packing dimensions. Let $K$ be a compact metric space and let us denote by $C(K,\mathbb{R}^n)$ the set of continuous maps from $K$ to $\mathbb{R}^n$ endowed with the maximum norm. Let $\dim_{*}$ be one of the topological dimension $\dim_T$, the Hausdorff dimension $\dim_H$, or the packing dimension $\dim_P$. Define $$d_{*}^n(K)=\inf\{\dim_{*}(K\setminus F): F\subset K \textrm{ is $σ$-compact with } \dim_T F<n\}.$$ We prove that $d^n_{*}(K)$ is the right notion to describe the dimensions of the fibers of a generic continuous map $f\in C(K,\mathbb{R}^n)$. In particular, we show that $\sup\{\dim_{*}f^{-1}(y): y\in \mathbb{R}^n\} =d^n_{*}(K)$ provided that $\dim_T K\geq n$, otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume $\dim_T K\geq n$. If $K$ is sufficiently homogeneous, then we can say much more. For example, we prove that $\dim_{*}f^{-1}(y)=d^n_{*}(K)$ for a generic $f\in C(K,\mathbb{R}^n)$ for all $y\in \textrm{int} f(K)$ if and only if $d^n_{*}(U)=d^n_{*}(K)$ or $\dim_T U<n$ for all open sets $U\subset K$. This is new even if $n=1$ and $\dim_{*}=\dim_H$. It is known that for a generic $f\in C(K,\mathbb{R}^n)$ the interior of $f(K)$ is not empty. We augment the above characterization by showing that $\dim_T \partial f(K)=\dim_H \partial f(K)=n-1$ for a generic $f\in C(K,\mathbb{R}^n)$.

math.CA↗

arXiv2016

Restrictions of Hölder continuous functions

For $0<α<1$ let $V(α)$ denote the supremum of the numbers $v$ such that every $α$-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \leq V(α)\leq 1/(2-α)$ and asked whether the upper bound is sharp. We show that in fact $V(α)=\max\{1/2,α\}$. Let $\dim_{H}$ and $\overline{\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(α)$ is a consequence of the following theorem. Let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. Then, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>\max\{1-α,α\}$ and $B\colon A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>1-α$ and $B\colon A\to \mathbb{R}$ is $β$-Hölder continuous for some $β>α$. The zero set and the set of record times of $B$ witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic $α$-Hölder continuous functions. Finally, let $\{\mathbf{B}(t): t\in [0,1]\}$ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set $D\subset [0,1]$ such that $\dim_{H} D\geq 1/3$ and $\mathbf{B}\colon D\to \mathbb{R}^2$ is non-decreasing in each coordinate. It remains open whether $1/3$ is best possible.

math.PR↗

arXiv2016

Increasing subsequences of random walks

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that $\mathbb{E} L=(2+o(1)) \sqrt{n}$. We consider the case when $\{S_i\}_{i\leq n}$ is a random walk on $\mathbb{R}$ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies $\mathbb{E} L\geq c\sqrt{n}$. Our main result is an upper bound $\mathbb{E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If $\{S_i\}_{i\leq n}$ is a simple random walk on $\mathbb{Z}$, we improve the lower bound by showing that $\mathbb{E} L \geq c\sqrt{n} \log{n}$. We also show that if $\{\mathbf{S}_i\}$ is a simple random walk in $\mathbb{Z}^2$, then there is a subsequence of $\{\mathbf{S}_i\}_{i\leq n}$ of expected length at least $cn^{1/3}$ that is increasing in each coordinate. The above one-dimensional result yields an upper bound of $n^{1/2 + o(1)}$. The problem of determining the correct exponent remains open.

math.PR↗

arXiv2016

Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $μ$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $μ\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{Haar ambivalent} if it is neither shy nor prevalent. The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $λ$ are singletons form a non-shy set in $C[0,1]$. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $λ\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.

math.CA↗