arXiv subjects

Balázs Szegedy

Publications and source records attributed to Balázs Szegedy.

32 records · Page 2Linked to original sources

arXiv2020

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer $k\geq 0$ the joint distribution's marginals on affine subcubes of dimension $k$ are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

math.DS↗

arXiv2019

On nilspace systems and their morphisms

A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace X equipped with a group of nilspace translations acting on X. Nilspace systems appear in different guises in several recent works, and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first and third named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors.

math.DS↗

arXiv2016

Measures on the square as sparse graph limits

We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on $[0,1]^2$. This leads to a generalization of dense graph limit theory to sparse graph sequences.

math.CO↗

arXiv2016

A continuous model for systems of complexity 2 on simple abelian groups

It is known that if $p$ is a sufficiently large prime then for every function $f:\mathbb{Z}_p\to [0,1]$ there exists a continuous function on the circle $f':\mathbb{T}\to [0,1]$ such that the averages of $f$ and $f'$ across any prescribed system of linear forms of complexity 1 differ by at most $ε$. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing $\mathbb{T}$ with the torus $\mathbb{T}^2$ equipped with a specific filtration. To this end we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps, and we combine this with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on $\mathbb{Z}_p$. For instance, let $m_4(α,\mathbb{Z}_p)$ denote the minimum, over all sets $A\subset \mathbb{Z}_p$ of cardinality at least $αp$, of the density of 4-term arithmetic progressions inside $A$. We show that $\lim_{p\to \infty} m_4(α,\mathbb{Z}_p)$ is equal to the infimum, over all measurable functions $f:\mathbb{T}^2\to [0,1]$ with $\int_{\mathbb{T}^2}f\geq α$, of the following integral: $$ \int_{\mathbb{T}^5} f\binom{x_1}{y_1}\; f\binom{x_1+x_2}{y_1+y_2}\; f\binom{x_1+2x_2}{y_1+2y_2+y_3}\; f\binom{x_1+3 x_2}{y_1+3y_2+3y_3} \,dμ_{\mathbb{T}^5}(x_1,x_2,y_1,y_2,y_3). $$

math.CO↗

arXiv2015

On large girth regular graphs and random processes on trees

We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random $d$-reguar graphs and ergodic theory on $T_d$. Using this correspondence principle together with entropy inequalities for typical processes we prove a family of combinatorial statements about random $d$-regular graphs.

math.PR↗

arXiv2015

On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Král', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemerédi's theorem: for any compact abelian group $G$, any measurable set $A\subset G$ with Haar probability $μ(A)\geqα>0$ satisfies $$\int_G\int_G\; 1_A\big(x\big)\; 1_A\big(x+r\big) \cdots 1_A\big(x+(k-1)r\big) \; dμ(x) dμ(r) \geq c,$$ where the constant $c=c(α,k)>0$ is valid uniformly for all $G$. This result is shown to hold more generally for any translation-invariant system of $r$ linear equations given by an integer matrix with coprime $r\times r$ minors.

math.CO↗

arXiv2014

The automorphism group of a graphon

We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on $k$-tuples of points. Among applications we study the graph algebras defined by finite rank graphons and the space of node-transitive graphons.

math.CO↗

arXiv2013

Graph properties, graph limits and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity.

math.CO↗

arXiv2013

Limits of local-global convergent graph sequences

The colored neighborhood metric for sparse graphs was introduced by Bollobás and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.

math.CO↗

arXiv2010

Limits of compact decorated graphs

Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an element of a compact second-countable Hausdorff space K. The "local structure" of these objects can be explored by a sampling process, which is shown to be equivalent to knowing homomorphism numbers from graphs whose edges are decorated by continuous functions on K. The model includes multigraphs with bounded edge multiplicities, graphs whose edges are weighted with real numbers from a finite interval, edge-colored graphs, and other models. In all these cases, a limit object can be defined in terms of 2-variable functions whose values are probability distributions on K.

math.CO↗

arXiv2010

The graph theoretic moment problem

We study an analogue of the classical moment problem in the framework where moments are indexed by graphs instead of natural numbers. We study limit objects of graph sequences where edges are labeled by elements of a topological space. Among other things we obtain strengthening and generalizations of the main results of previous papers characterizing reflection positive graph parameters, graph homomorphism numbers, and limits of simple graph sequences. We study a new class of reflection positive partition functions which generalize the node-coloring models (homomorphisms into weighted graphs).

math.CO↗

arXiv2010

Regularity partitions and the topology of graphons

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.

math.CO↗

arXiv2009

Random Graphons and a Weak Positivstellensatz for Graphs

In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this paper we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: We show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

math.CO↗

arXiv2008

A measure-theoretic approach to the theory of dense hypergraphs

In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new results including a strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also prove the equivalence of various notions for convergence of hypergraphs and we construct limit objects for such sequences. We prove that the limit objects are unique up to a certain family of measure preserving transformations. As our main tool we study the integral and measure theory on the ultraproduct of finite measure spaces which is interesting on its own right.

math.CO↗