Showing papers in "Israel Journal of Mathematics in 2014"

Model theory of operator algebras II: model theory

Abstract: We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.

Strongly dependent theories

Abstract: We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.

Smooth Fréchet globalizations of Harish-Chandra modules

Abstract: We give an alternative proof of the Casselman-Wallach globalization theorem. The approach is based on lower bounds for matrix coefficients on a reductive group.

On the KŁR conjecture in random graphs

Abstract: The KŁR conjecture of Kohayakawa, Łuczak, and Rodl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rodl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.

Convex equipartitions via Equivariant Obstruction Theory

Abstract: We describe a regular cell complex model for the configuration space F(ℝ d , n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

Mean dimension and an embedding problem: An example

Abstract: For any positive integer D, we construct a minimal dynamical system with mean dimension equal to D/2 that cannot be embedded into (([0, 1] D )ℤ, shift).

Homology of homogeneous divisors

Abstract: This work deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors not coming from either hyperplane arrangements or discriminants in singularity theory.

On the best constants in the Khintchine inequality for Steinhaus variables

Abstract: Let Sj: (Ω, P) → S1 ⊂ ℂ be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S1. We determine the best constants ap in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern- ulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p0 }~ 0.4756. For p ≥ 1 the best constants ap have been known for some time. The result implies for a norm 1 sequence x ∈ ℂn, ‖x‖2 = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.

Overconvergent modular sheaves and modular forms for GL 2/F

Abstract: Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ℚ, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.

Primitive words, free factors and measure preservation

Abstract: Let Fk be the free group on k generators. A word w ∈ Fk is called primitive if it belongs to some basis of Fk. We investigate two criteria for primitivity, and consider more generally subgroups of Fk which are free factors.

On toral eigenfunctions and the random wave model

Abstract: The purpose of this Note is to provide a deterministic implementation of the random wave model for the number of nodal domains in the context of the two-dimensional torus. The approach is based on recent work due to Nazarov and Sodin and arithmetical properties of lattice points on circles.

Dilation of Ritt operators on L p -spaces

Abstract: For any Ritt operator T: L p (Ω) → L p (Ω), for any positive real number α, and for any x ∈ L p (Ω), we consider $${\left\| x \right\|_{T,\alpha }} = {\left\| {{{\left( {\sum\limits_{k = 1}^\infty {{k^{2\alpha - 1}}} {{\left| {{T^{k - 1}}{{(I - T)}^\alpha }x} \right|}^2}} \right)}^{\frac{1}{2}}}} \right\|_{{L^p}}}$$ . We show that if T is actually an R-Ritt operator, then the square functions \({\left\| {} \right\|_{T,\alpha }}\) are pairwise equivalent. Then we show that T and its adjoint T*: L p′ (Ω) → L p′ (Ω) both satisfy uniform estimates \({\left\| x \right\|_{T,1}} \leqslant {\left\| x \right\|_{{L^p}}}\) and \({\left\| y \right\|_{T*,1}} \leqslant {\left\| y \right\|_{{L^{p'}}}}\) for x ∈ L p (Ω) and y ∈ L p′ (Ω) if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space \(\tilde \Omega \), an isomorphism \(U:{L^p}\tilde \Omega \to {L^p}\tilde \Omega \) such that {U n : n ∈ ℤ} is bounded, as well as two bounded maps \({L^p}(\Omega )\buildrel J \over \longrightarrow {L^p}(\tilde \Omega )\buildrel Q \over \longrightarrow {L^p}(\Omega )\) such that T n = QU n J for any n ≥ 0. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L p -spaces.

Geometry and entropy of generalized rotation sets

Abstract: For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (ϕ1, ..., ϕ m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ℝ m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ℝ m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot Pt (Φ). Finally, we study the entropy function w ↦ H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ↦ H(w) is real-analytic in the interior of the rotation set.

Counting sum-free sets in abelian groups

Abstract: In this paper we study sum-free sets of order m in finite abelian groups. We prove a general theorem about independent sets in 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order m in abelian groups G whose order n is divisible by a prime q with q ≡ 2 (mod 3), for every m ⩾ \(C(q)\sqrt {n\log n} \), thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sumfree subsets of size m are contained in a maximum-size sum-free subset of G. We also give a completely self-contained proof of this statement for abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of a fixed size in an (n, d, λ)-graph.

Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries

Abstract: We consider a semilinear Dirichlet problem with an unbounded and indefinite potential and a superlinear reaction which need not satisfy the usual, in such cases, Ambrosetti-Rabinowitz condition. Using a combination of variational methods (critical point theory) with truncation and comparison techniques, with Morse theory and with flow invariance arguments, we show that the problem has at least seven nontrivial smooth solutions and provide sign information for all of them.

Kato’s Euler system and rational points on elliptic curves I: A p -adic Beilinson formula

Abstract: This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic etale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.

On the Hardy-Littlewood maximal function for the cube

Abstract: It is shown that the Hardy-Littlewood maximal function associated to the cube in ℝ n obeys dimensional free bounds in L p for p > 1. Earlier work only covered the range p > $$\frac{3}{2}$$ .

The isotropic position and the reverse Santaló inequality

Abstract: We present proofs of the reverse Santalo inequality, the existence of M- ellipsoids and the reverse Brunn-Minkowski inequality, using purely con- vex geometric tools. Our approach is based on properties of the isotropic position.

Minimal functions on the random graph

Abstract: We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we rederive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the maximal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.

On gradient Ricci solitons conformal to a pseudo-Euclidean space

Abstract: We consider gradient Ricci solitons, conformal to an n-dimensional pseudo-Euclidean space, which are invariant under the action of an (n − 1)-dimensional translation group. We provide all such solutions in the case of steady gradient Ricci solitons.

Foundations for an iteration theory of entire quasiregular maps

Abstract: The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.

Effective estimates on indefinite ternary forms

Abstract: We give an effective proof of a theorem of Dani and Margulis regarding values of indefinite ternary quadratic forms at primitive integer vectors. The proof uses an effective density-type result for orbits of the groups SO(2, 1) on SL(3, ℝ)/SL(3, ℤ).

On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis

Abstract: In this paper we establish an estimate for the rate of convergence of the Krasnosel’skiĭ-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffdingtype inequality to majorize the expected value of a convex function of these sums using Poisson distributions.

Noise sensitivity in continuum percolation

Abstract: We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability pc not equal 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with pc bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

Yet another short proof of Bourgain’s distortion estimate for embedding of trees into uniformly convex Banach spaces

Abstract: We use a self-improvement argument to give a very short and elementary proof of the result of Bourgain saying that infinite regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.

Base sizes for S -actions of finite classical groups

Abstract: Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.

Bernoulli actions of sofic groups have completely positive entropy

Abstract: We prove that every Bernoulli action of a sofic group has completely positive entropy with respect to every sofic approximation net. We also prove that every Bernoulli action of a finitely generated free group has the property that each of its nontrivial factors with a finite generating partition has positive f-invariant.

Unconditional structures of translates for L p (ℝ d )

Abstract: We prove that a sequence (f i ) =1 ∞ of translates of a fixed f ∈ L p (ℝ) cannot be an unconditional basis of L p (ℝ) for any 1 ≤ p < ∞. In contrast to this, for every 2 < p < ∞, d ∈ ℕ and unbounded sequence (λ n ) n∈ℕ ⊂ ℝ d we establish the existence of a function f ∈ L p (ℝ d ) and sequence (g n *) n∈ℕ ⊂ L p *(ℝ d ) such that $${({T_{{\lambda _n}}}f,g_n^*)_{n \in {\Bbb N}}}$$ forms an unconditional Schauder frame for L p (ℝ d ). In particular, there exists a Schauder frame of integer translates for L p (ℝ) if (and only if) 2 < p < ∞.

Puiseux series dynamics of quadratic rational maps

Abstract: We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.

On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel

Abstract: We show that the Maharam extension of a type III, conservative and nonsingular K Bernoulli is a K-transformation. This together with the fact that the Maharam extension of a conservative transformation is conservative gives a negative answer to Krengel’s and Weiss’s questions about existence of a type II∞ or type IIIλ with λ ≠ 1 Bernoulli shift. A conservative non-singular K, in the sense of Silva and Thieullen, Bernoulli shift is either of type II1 or of type III1.