Borel Sets Research Articles

Pairs in discrete lattice orbits with applications to Veech surfaces (with an appendix by Jon Chaika)

Let \Lambda_{1} , \Lambda_{2} be two discrete orbits under the linear action of a lattice \Gamma<\mathrm{SL}_{2}(\mathbf{R}) on the Euclidean plane. We prove a Siegel–Veech-type integral formula for the averages \sum_{{\bf x}\in\Lambda_1}\sum_{{\bf y}\in\Lambda_2}f({\bf x},{\bf y}), from which we derive new results for the set S_{M} of holonomy vectors of saddle connections of a Veech surface M . This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in S_{M} with bounded determinant and on the number of pairs in S_{M} with bounded distance. This last estimate is used in the appendix to prove that for almost every (\theta,\psi)\in S^{1}\times S^{1} the translation flows F_{\theta}^{t} and F_{\psi}^{t} on any Veech surface M are disjoint.

Regularity of the minmax value and equilibria in multiplayer Blackwell games

AbstractA real-valued function φ that is defined over all Borel sets of a topological space is regular if for every Borel set W, φ(W) is the supremum of φ(C), over all closed sets C that are contained in W, and the infimum of φ(O), over all open sets O that contain W.We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player’s minmax value is regular.We then use the regularity of the minmax value to establish the existence of ε-equilibria in two distinct classes of Blackwell games. One is the class of n-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n − 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

In this paper, we study the heat content for the Schrödinger operator related to a subordinate Brownian motion and we also establish its small time asymptotic behavior for suitable potentials V. The case V=c1Ω for c>0 and Ω a Borel set of finite measure is investigated in detail.

Sequences with increasing subsequence

We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.

Measure theoretic aspects of the finite Hilbert transform

AbstractThe finite Hilbert transform , when acting in the classical Zygmund space (over ), was intensively studied in [8]. In this note, an integral representation of is established via the ‐valued measure for each Borel set . This integral representation, together with various non‐trivial properties of , allows the use of measure theoretic methods (not available in [8]) to establish new properties of . For instance, as an operator between Banach function spaces is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for plays a crucial role.

Anti-classification results for weakly mixing diffeomorphisms

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.

On the first Banach problem concerning condensations of absolute $$\kappa$$-Borel sets onto compacta

On the first Banach problem concerning condensations of absolute $$\kappa$$-Borel sets onto compacta

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

How much can heavy lines cover?

AbstractOne formulation of Marstrand's slicing theorem is the following. Assume that , and is a Borel set with . Then, for almost all directions , almost all of is covered by lines parallel to with . We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction , is it true that a strictly less than ‐dimensional part of is covered by the heavy lines , namely those with ? A positive answer for ‐regular sets was previously obtained by the first author. The answer for general Borel sets turns out to be negative for and positive for . More precisely, the heavy lines can cover up to a dimensional part of in a generic direction. We also consider the part of covered by the ‐heavy lines, namely those with for . We establish a sharp answer to the question: how much can the ‐heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub‐uniformly distributed sets, which generalise Ahlfors‐regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors‐regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors‐regular sets to the class of sub‐uniformly distributed sets.

Exponential Iteration and Borel Sets

Exponential Iteration and Borel Sets

Limit theorems for a branching random walk in a random or varying environment

We consider a branching random walk on the real line with a stationary and ergodic environment (ξn) indexed by time, in which a particle of generation n gives birth to a random number of particles of the next generation, which move on the real line; the joint distribution of the number of children and their displacements on the real line depends on the environment ξn at time n. Let Zn be the counting measure at time n, which counts the number of particles of generation n situated in a Borel set of the real line. For the case where the corresponding branching process is supercritical, we establish limit theorems such as large and moderate deviation principles, central and local limit theorems on the counting measures Zn, convergence of the free energy, law of large numbers on the leftmost and rightmost positions at time n, and the convergence to infinite divisible laws. The varying environment case is also considered.

On the Hausdorff dimension of radial slices

Let \(t \in (1,2)\), and let \(B \subset \mathbb{R}^{2}\) be a Borel set with \(\dim_{\mathrm{H}} B > t\). I show that \(\mathcal{H}^{1}(\{e \in S^{1} \colon \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0\) for all \(x \in \mathbb{R}^{2} \setminus E\), where \(\dim_{\mathrm{H}} E \leq 2 - t\). This is the sharp bound for \(\dim_{\mathrm{H}} E\). The main technical tool is an incidence inequality of the form \(\mathcal{I}_{\delta}(\mu,\nu) \lesssim_{t} \delta \cdot \sqrt{I_{t}(\mu)I_{3 - t}(\nu)}\), \(t \in (1,2)\), where \(\mu\) is a Borel measure on \(\mathbb{R}^{2}\), and \(\nu\) is a Borel measure on the set of lines in \(\mathbb{R}^{2}\), and \(\mathcal{I}_{\delta}(\mu,\nu)\) measures the \(\delta\)-incidences between \(\mu\) and the lines parametrised by \(\nu\). This inequality can be viewed as a \(\delta^{-\epsilon}\)-free version of a recent incidence theorem due to Fu and Ren. The proof in this paper avoids the high-low method, and the induction-on-scales scheme responsible for the \(\delta^{-\epsilon}\)-factor in Fu and Ren's work. Instead, the inequality is deduced from the classical smoothing properties of the \(X\)-ray transform.

Riemann Integral on Fractal Structures

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ-measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of Rn meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets.

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

Abstract We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

Kolmogorov extension theorem for non-probability measures on Cayley trees

In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a [Formula: see text]-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.

Convex weak concordance measures and their constructions

Considering the framework of weak concordance measures introduced by Liebscher in 2014, we propose and study convex weak concordance measures. This class of dependence measures contains as a proper subclass the class of all convex concordance measures, introduced and studied by Mesiar et al. in 2022, and thus it also covers the well-known concordance measures as Spearman's ρ, Gini's γ and Blomqvist's β. The class of all convex weak concordance measures also contains, for example, Spearman's footrule ϕ, which is not a concordance measure. In this paper, we first introduce basic convex weak concordance measures built in general by means of a single point (u,v)∈▽={(u,v)∈]0,1[2|u≥v} and its transpose (v,u) only. Then, based on basic convex weak concordance measures and probability measures on the Borel subsets of ▽, two rather general constructions of convex weak concordance measures are proposed, discussed and exemplified. Inspired by Edwards et al., probability measures-based constructions are generalized to Borel measures on B(]0,1[2)-based constructions also allowing some infinite measures to be considered. Finally, it is shown that the presented constructions also cover the mentioned standard (convex weak) concordance measures ρ, γ, β, ϕ and provide alternative formulas for them.

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d([γ],[σ]):=||γ⁎−σ⁎||1 using the (constant-speed) tree-reduced representatives γ⁎ and σ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24].

The Borel partition spectrum at successors of singular cardinals

Assuming that 0 † 0^\dagger does not exist, we prove that if there is a partition of R \mathbb R into ℵ ω \aleph _\omega Borel sets, then there is also a partition of R \mathbb R into ℵ ω + 1 \aleph _{\omega +1} Borel sets.

An Exceptional Set Estimate for Restricted Projections to Lines in mathbb R^3

Let gamma :[0,1]rightarrow mathbb S^{2} be a non-degenerate curve in mathbb R^3, that is to say, det big (gamma (theta ),gamma '(theta ),gamma ''(theta )big )ne 0. For each theta in [0,1], let l_theta =text {span}(gamma (theta )) and rho _theta :mathbb R^3rightarrow l_theta be the orthogonal projections. We prove an exceptional set estimate. For any Borel set Asubset mathbb R^3 and 0le sle 1, define E_s(A):={theta in [0,1]: dim (rho _theta (A))<s}. We have dim (E_s(A))le max {0,1+frac{s-dim (A)}{2}}.

Congruence Representations via Soft Ideals in Soft Topological Spaces

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft σ-algebra. In this work, we mainly focus on the soft σ-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft σ-algebra of soft sets with the Baire property includes the soft Borel σ-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one.