arXiv subjects

Jan Grebík

Publications and source records attributed to Jan Grebík.

At least 19 recordsLinked to original sources

arXiv2026

Borel dimension growth and hyperfiniteness

We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^γ))$ is hyperfinite, where $γ\approx 0.1523$ is the unique real root of $(1 - γ)^3 - 4γ= 0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp(O(r^γ))$ for $γ$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.

math.GR↗

arXiv2026

Fractional Divisibility of Spheres with Partially Generic Sets of Rotations

We say that an r-tuple $(g_1,...,g_r)$ of special orthogonal $d\times d$ matrices fractionally divides the $(d-1)$-dimensional sphere $S$ if there is a non-constant function in $L^2(S)$ such that its translations by $g_1,...,g_r$ sum up to the constant-1 function. Our main result shows, informally speaking, that fractional divisibility is impossible if at least $r/2$ rotations are ``generic".

math.MG↗

arXiv2026

Bolzano: Case Studies in LLM-Assisted Mathematical Research

We report new results on eight problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orchestrates rounds of interaction between parallel prover agents and a verifier agent while maintaining a persistent knowledge base that is carried across rounds. Classified using the significance-autonomy taxonomy of Feng et al., six of the eight results reach the level of publishable research, and five of the eight were produced essentially autonomously by Bolzano. Our results provide evidence that LLMs can contribute meaningfully to mathematical research, complementing recent reports by Bubeck et al., Woodruff et al., and others.

cs.CL↗

arXiv2026

Complexity of Finite Borel Asymptotic Dimension

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbfΣ^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

math.LO↗

arXiv2026

Convergence of spectra of digraph limits

The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.

math.CO↗

arXiv2025

Deterministic Distributed Algorithms and Measurable Combinatorics on $Δ$-Regular Forests

We investigate the connections between the fields of distributed computing and measurable combinatorics by considering complexity classes of locally checkable labeling problems on regular forests. We show that the most important deterministic complexity classes from the LOCAL model of distributed computing exactly coincide with well-studied classes in measurable combinatorics. Namely, first we show that a locally checkable labeling problem admits a continuous solution if and only if it can be solved by a deterministic local algorithm with complexity $O(\log^* n)$. Second, our main result states that, surprisingly, a locally checkable labeling problem admits a Baire measurable solution if and only if it can be solved by a local algorithm with complexity $O(\log n)$. These theorems suggest the existence of deeper connections between the two frameworks. Furthermore, the latter result relies on a complete combinatorial characterization of the classes in question, and as a by-product, it shows that membership in these classes is decidable.

math.LO↗

arXiv2025

Vanishing uniqueness thresholds in Voronoi percolation on products

We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(λ)$ at small intensities $λ>0$ based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge2,d\ge3$ and products of hyperbolic spaces $\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}$ for $k\ge2, d_i\ge2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.

math.PR↗

arXiv2025

Large deviation principles for graphon sampling

We investigate possible large deviation principles (LDPs) for the $n$-vertex sampling from a given graphon with various speeds $s(n)$ and resolve all the cases except when the speed $s(n)$ is of order $n^2$. For quadratic speed $s=(c+o(1))n^2$, we establish an LDP for an arbitrary $k$-step graphon, which extends a result of Chatterjee and Varadhan [Europ. J. Combin., 32 (2011) 1000-1017] who did this for $k=1$ (that is, for the homogeneous binomial random graphs). This is done by reducing the problem to the LDP for stochastic $k$-block models established recently by Borgs, Chayes, Gaudio, Petti and Sen ["A large deviation principle for block models", arxiv:2007.14508, 2020]. Also, we improve some results by Borgs et al.

math.PR↗

arXiv2025

Poisson-Voronoi percolation in higher rank

We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally different from situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (arXiv:2307.01194) and provides an alternative proof strategy for Gaboriau's fixed price problem. As a further application of our result, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and arbitrarily small expected degree, answering a question inspired by Hutchcroft-Pete (Invent. math. 221 (2020)).}

math.PR↗

arXiv2025

From descriptive to distributed

In the past couple of years a rich connection has been found between the fields of descriptive set theory and distributed computing. Frequently, and less surprisingly, finitary algorithms can be adopted to the infinite setting, resulting in theorems about infinite, definable graphs. In this survey, we take a different perspective and illustrate how results and ideas from descriptive set theory provide new insights and techniques to the theory of distributed computing. We focus on the two classical topics from graph theory, vertex and edge colorings. After summarizing the up-to-date results from both areas, we discuss the adaptation of Marks' games method to the LOCAL model of distributed computing and the development of the multi-step Vizing's chain technique, which led to the construction of the first non-trivial distributed algorithms for Vizing colorings. We provide a list of related open problems to complement our discussion. Finally, we describe an efficient deterministic distributed algorithm for Brooks coloring on graphs of subexponential growth.

math.LO↗

arXiv2025

Complexity of Linear Equations and Infinite Gadgets

We investigate the descriptive set-theoretic complexity of the solvability of a Borel family of linear equations over a finite field. Answering a question of Thornton, we show that this problem is already hard, namely $Σ^1_2$-complete. This implies that the split between easy and hard problems is at a different place in the Borel setting than in the case of the CSP Dichotomy.

math.LO↗

arXiv2024

Periodicity and decidability of translational tilings by rational polygonal sets

The periodic tiling conjecture asserts that if a region $Σ\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in $\mathbb R$, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.

math.CO↗

arXiv2024

Measurable Vizing's theorem

We prove a full measurable version of Vizing's theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree uniformly bounded by $Δ\in \mathbb{N}$ defined on a standard probability space $(X,μ)$ admits a $μ$-measurable proper edge coloring with $(Δ+1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)] who derived the same conclusion under the additional assumption that the measure $μ$ is $\mathcal{G}$-invariant.

math.LO↗

arXiv2024

On Homomorphism Graphs

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $Δ>2$ it is impossible to give a simple characterization of acyclic $Δ$-regular Borel graphs with Borel chromatic number at most $Δ$: such graphs form a $\mathbfΣ^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

math.LO↗

arXiv2023

Large Deviation Principles for Block and Step Graphon Random Graph Models

Borgs, Chayes, Gaudio, Petti and Sen [arXiv:2007.14508] proved a large deviation principle for block model random graphs with rational block ratios. We strengthen their result by allowing any block ratios (and also establish a simpler formula for the rate function). We apply the new result to derive a large deviation principle for graph sampling from any given step graphon.

math.PR↗

arXiv2023

Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+] .

math.CO↗

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↗