Zero-one Law Research Articles

Almost Everywhere Convergence Questions of Series of Translates of Non-Negative Functions

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{{\infty}}f(nx)$. A more general, additive version of this problem is the following: Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: \mathbb R\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= \mathbb R$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= \mathbb R$ modulo sets of measure zero. If $ {\Lambda}$ is not of type 1 we say that $ {\Lambda}$ is of type 2. The exact characterization of type $1$ and type $2$ sets is still not known. The part of the paper discussing results concerning this question is based on several joint papers written at the beginning with J-P. Kahane and D. Mauldin, later with B. Hanson, B. Maga and G. Vértesy. Apart from results from the above project we also cover historic background, other related results and open questions.

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.

Dynamical Gibbs–non-Gibbs transitions in Widom–Rowlinson models on trees

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order $d$ (which has $d + 1$ nearest neighbours), depending on repulsion strength $\beta$ between particles of different signs and on an activity parameter $\lambda$ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength $\beta$. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any $d \geq 2$. In our second and main result, we also show that for large $\beta$ and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity $\lambda$ increases, assuming that $d \geq 4$. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.

Simultaneous p-adic Diophantine approximation

AbstractThe aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

Geometric progressions in distance spaces; applications to fixed points and coincidence points

Conditions on spaces $X$ with generalized distance $\rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $&lt;1$ (that is, each sequence $\{ x_i\}\subset X$ satisfying $\rho_X(x_{i+1},x_i)\leq \gamma \rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $\gamma &lt; 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $\rho_X$ in it satisfies $\rho_X(x,z) \leq \rho_X(x,y)+(\rho_X(y,z))^\eta$, $x,y,z \in X$, for some $\eta\in (0,1)$, that is, if the function $f\colon\mathbb{R}_+^{2} \to \mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{\eta}$. Next, for $f(r_1,r_2)=\max\{ r_1^{\eta}, r_2^{\eta} \}$, where $\eta \in (0,2^{-1}]$, it is shown that for any $\gamma &gt; 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $\gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $&lt;1$ is a Cauchy sequence or, for any $\gamma\in (0,1)$, there exists a geometric progression with ratio $\gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.

A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group

Nous prouvons une loi zéro-un pour la mesure stationnaire pour des ensembles algébriques en généralisant les résultats de Furstenberg (Proc. Sympos. Pure Math. 26 (1973) 193–229) et Guivarc’h et Le Page (Ann. Inst. Henri Poincaré Probab. Stat. 52(2) (2016) 503–574). Comme application, nous établissons un théorème local limite pour les coefficients de marches aléatoires sur le groupe linéaire général.

Modified shrinking target problems on self-conformal sets

In this paper, we investigate a modified version of the shrinking target problem on self-conformal sets, which unifies the shrinking target problems and quantitative recurrence properties. Let J be a self-conformal set generated by a conformal iterated function system satisfying the open set condition, and let T:J→J be the expanding map induced by the left shift. We will study the size of the following set:R(f,φ):={x∈J:|Tnx−f(x)|<φ(n)for infinitely manyn∈N}, where f:J→J is a Lipschitz function and φ:N→R+ is a positive function defined on N. The Hausdorff dimension and zero-one law on the μ-measure of R(f,φ) are completely obtained, where μ stands for the natural self-conformal measure supported on J.

A Zero-One Law for the Rates of Convergence in the Birkhoff Ergodic Theorem with Continuous Time

A Zero-One Law for the Rates of Convergence in the Birkhoff Ergodic Theorem with Continuous Time

Tail algebras for monotone and q-deformed exchangeable stochastic processes

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti’s theorem for this type of processes. In addition, since the vacuum state on the q-deformed \(C^*\)-algebra is the only exchangeable state when \(|q|<1\), we draw our attention to its tail algebra, which turns out to obey a zero-one law.

Singular integrals of subordinators with applications to structural properties of SPDEs

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution Z t Z_t , (2) a small ball probability of Z t Z_t , (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.

Strictly balanced uniform hypergraphs and generalizations of Zero-One Law

In this work we describe the spectra of all rational numbers that could be a density of a strictly balanced uniform hypergraph. We also introduce some specific constructions of strictly balanced uniform hypergraphs, and exploit them to generalize some results about Zero-One Law and Zero-One k-Law to the case of random uniform hypergraphs.

Geodesics and metric ball boundaries in Liouville quantum gravity

Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $$\gamma \in (0,2)$$ . We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (Commun Math Phys 378(1):625–689, 2020. arXiv:1909.08588 ) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.

Zero-One Laws for Existential First-Order Sentences of Bounded Quantifier Depth

For any fixed positive integer k , let α k denote the smallest α ∈ (0,1) such that the random graph sequence { G ( n, n -α )} n does not satisfy the zero-one law for the set ε k of all existential first-order sentences that are of quantifier depth at most k . This article finds upper and lower bounds on α k , showing that as k → ∞, we have α k = ( k - 2 - t ( k )) -1 for some function t ( k ) = Θ ( k -2 ). We also establish the precise value of α k when k = 4.

Zero-one law for the rates of convergence in the Birkhoff ergodic theorem with continuous time

Zero-one law for the rates of convergence in the Birkhoff ergodic theorem with continuous time

A zero-one law for Markov chains

We prove an analogue of the classical zero-one law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event A from the tail σ-algebra of MC , for large n, with probability near one, the trajectories of the MC are in states i, where is either near 0 or near 1. A similar statement holds for the entrance σ-algebra when n tends to . To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by or in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.

The almost sure theory of finite metric spaces

We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $\sigma$ in the language of pure metric spaces and any $\epsilon>0$, the probability that the difference of the value of $\sigma$ in a random metric space of size $n$ and the value of $\sigma$ in any model of $T_{\mathrm{as}}$ is less than $\epsilon$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.

On the Minimum Node Degree and k-Connectivity in Inhomogeneous Random K-Out Graphs

Inhomogeneous random K-out graphs were recently introduced to model heterogeneous sensor networks secured by random pairwise key predistribution schemes. First, each of the n nodes is classified as type-1 (respectively, type-2) with probability 0 <; μ <; 1 (respectively, 1-μ) independently from each other. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> arcs towards K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> distinct nodes) selected uniformly at random, and then the orientation of the arcs is ignored. It was recently established that the inhomogeneous random K-out graph is 1-connected asymptotically almost surely (a.a.s.) if and only if K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> =ω(1). In this work, we analyze the k-connectivity of inhomogeneous random K-out graphs; i.e., with k=1, 2, <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> ⋯</sub> , the property that the network remains connected despite the removal of any k-1 nodes or links. We first establish a zero-one law for the property that the minimum node degree is at least k. In particular, we present scaling conditions on μ and K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> such that the resulting graph has minimum degree at least k with probability approaching one (respectively, zero) constituting the one-law (respectively, zero-law), as the number of nodes gets large. We show that for k=2, 3, <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋯</sub> , we need to set K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> = \frac 11-μ(logn +(k-2)loglogn + ω(1)) for the network to have a minimum node degree of at least k a.a.s. Next, we prove that having K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> = \frac 11-μ(logn +(k-2)loglogn + ω(1)) also ensures that the graph is k-connected a.a.s., meaning that the zero-one laws for minimum node degree and k-connectivity coincide. We present simulation results to demonstrate the usefulness of the results in the finite node regime. The results given here indicate an interesting fact about inhomogeneous random K-out graphs, i.e., that the number of additional edges needed to go from 1-connectivity to k-connectivity with k ≥ 2 is unexpectedly larger as compared to many classical random graph models studied before.

A topological zero-one law and elementary equivalence of finitely generated groups

Let G denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces S⊆G satisfying the following zero-one law: for any sentence σ in the infinitary logic Lω1,ω, the set of all models of σ in S is either meager or comeager. In particular, we prove that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we show that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.

On limit points of spectra of first-order sentences with quantifier depth 4

We study the asymptotic behavior of probabilities of first-order properties of sparse binomial random graphs. We consider properties with quantifier depth not more than 4. It is known that the only possible limit points of the spectrum (i.e., the set of all positive α such that G(n,n−α) does not obey the zero-one law with respect to the property) of such a property are 1/2 and 3/5. We prove that 1/2 is not a limit point of the spectrum.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 &amp;lt; \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.