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Jan Grebík

Publications and source records attributed to Jan Grebík.

35 records · Page 2Linked to original sources
arXiv2023

Measurable tilings by abelian group actions

Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).

math.DS↗
arXiv2022

Divisibility of Spheres with Measurable Pieces

For an $r$-tuple $(γ_1,\ldots,γ_r)$ of special orthogonal $d\times d$ matrices, we say that the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is $(γ_1,\ldots,γ_r)$-divisible if there is a subset $A\subseteq S^{d-1}$ such that its translations by the rotations $γ_1,\ldots,γ_r$ partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set $A$ has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) $r$-tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" $r$-tuple, divisibility is possible with parts that have the property of Baire.

math.MG↗
arXiv2021

Cut distance identifying graphon parameters over weak* limits

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology (when considering the predual of $L^{1}$-functions). Doležal and Hladký [J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from homomorphism densities, $t(H,\cdot)$. This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kráľ, Martins, Pach and Wrochna [J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the «pumping property», and thus can be used in the proof of the Frieze-Kannan regularity lemma.

math.CO↗
arXiv2021

Borel equivalence relations induced by actions of tsi Polish groups

We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by countable structures but it is not essentially countable, then the equivalence relation ${\mathbb E}^{\mathbb N}_0=\mathbb E_3$ Borel reduces to it.

math.LO↗
arXiv2021

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

We study connections between distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics in the context of regular trees. We extend the Borel determinacy technique of Marks coming from descriptive combinatorics and adapt it to the area of distributed computing. Using this technique, we prove deterministic distributed $Ω(\log n)$-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest. We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity $O(\log n)$, extending the classification of Baire measurable colorings of Bernshteyn. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in $O(\log n)$ rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn -- arXiv 2004.04905].

math.CO↗
arXiv2021

Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics

We classify which local problems with inputs on oriented paths have so-called Borel solution and show that this class of problems remains the same if we instead require a measurable solution, a factor of iid solution, or a solution with the property of Baire. Together with the work from the field of distributed computing [Balliu et al. PODC 2019], the work from the field of descriptive combinatorics [Gao et al. arXiv:1803.03872, Bernshteyn arXiv:2004.04905] and the work from the field of random processes [Holroyd et al. Annals of Prob. 2017, Grebík, Rozhoň arXiv:2103.08394], this finishes the classification of local problems with inputs on oriented paths using complexity classes from these three fields. A simple picture emerges: there are four classes of local problems and most classes have natural definitions in all three fields. Moreover, we now know that randomness does \emph{not} help with solving local problems on oriented paths.

math.CO↗
arXiv2021

Fractional Isomorphism of Graphons

We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is characterized in terms of distributions on iterated degree measures, Markov operators, weak isomorphism of a conditional expectation with respect to invariant sub-$σ$-algebras and isomorphism of certain quotients of given graphons.

math.CO↗
arXiv2020

Tall $F_σ$ subideals of tall analytic ideals

Answering a question of Hrušák, we show that every analytic tall ideal on $ω$ contains an $F_σ$ tall ideal. We also give an example of an $F_σ$ tall ideal without a Borel selector.

math.LO↗
arXiv2020

Measurable versions of Vizing's theorem

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $Δ$ and $π$. The ``approximate'' version states that, for any Borel probability measure on the edge set and any $ε>0$, we can properly colour all but $ε$-fraction of edges with $Δ+π$ colours in a Borel way. The ``measurable'' version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most $Δ+π$ colours.

math.CO↗
arXiv2020

Relating the cut distance and the weak* topology for graphons

The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of $L^{1}$-functions). We prove that a sequence $W_1,W_2,W_3,\ldots$ of graphons converges in the cut distance if and only if we have equality of the sets of weak* accumulation points and of weak* limit points of all sequences of graphons $W_1',W_2',W_3',\ldots$ that are weakly isomorphic to $W_1,W_2,W_3,\ldots$. We further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above. This in particular provides an alternative proof of the theorem of Lovász and Szegedy about compactness of the space of graphons. We connect these results to "multiway cut" characterization of cut distance convergence from [Ann. of Math. (2) 176 (2012), no. 1, 151-219]. These results are more naturally phrased in the Vietoris hyperspace $K$ over graphons with the weak* topology. We show that graphons with the cut distance topology are homeomorphic to a closed subset of $K$, and deduce several consequences of this fact. From these concepts a new order on the space of graphons emerges. This order allows to compare how structured two graphons are. We establish basic properties of this "structurdness order".

math.CO↗
arXiv2019

Free sequences in P(ω)/fin

We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(ω)/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of Monk, we demonstrate the consistency of $ω_1 = \mathfrak{i} = \mathfrak{f} < \mathfrak{u} = ω_2$. In fact, this consistency is demonstrated in the model of S. Shelah for $\mathfrak{i} < \mathfrak{u}$. Our paper provides a streamlined and mostly self contained presentation of this construction.

math.LO↗
arXiv2018

A rigid Urysohn-like metric space

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well-known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $ω_1$ which has the same extension property as $R$, yet its group of automorphisms is trvial. We also present a similar, although technically more complicated, construction of a complete metric space of density $ω_1$, having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.

math.LO↗
arXiv2016

The Ascoli property for function spaces

The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $κ$-Fréchet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $κ$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stratifiable, then $C_p(X)$ is an Ascoli space. Consequently: (a) If $X$ is a complete metrizable space, then $C_p(X)$ is Ascoli iff $X$ is scattered. (b) If $X$ is a Čech-complete Lindelöf space, then $C_p(X)$ is Ascoli iff $X$ is scattered iff $C_p(X)$ is Fréchet-Urysohn. Moreover, we prove that for a paracompact space $X$ of point-countable type the following conditions are equivalent: (i) $X$ is locally compact. (ii) $C_k(X)$ is a $k_{\mathbb R}$-space. (iii) $C_k(X)$ is an Ascoli space. The Asoli spaces $C_k(X,[0,1])$ are also studied.

math.GN↗
arXiv2016

An example of a Fraïssé class without a Katětov functor

We disprove a conjecture from [W. Kubiś, D. Mašulović, Katětov functors, preprint, http://arxiv.org/abs/1412.1850] by showing the existence of a Fraïssé class $\mathcal{C}$ which does not admit a Katětov functor. On the other hand, we show that the automorphism group of the Fraïssé limit of $\mathcal{C}$ is universal, as it happens in the presence of a Katětov functor.

math.LO↗