Showing papers in "Israel Journal of Mathematics in 2016"

Optimal hardy-littlewood type inequalities for polynomials and multilinear operators

Abstract: In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust-Hilletype inequalities.

Determining Fuchsian groups by their finite quotients

Abstract: Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.

The weakness of being cohesive, thin or free in reverse mathematics

Abstract: Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey’s theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets.

A problem of Erdős and Sós on 3-graphs

Abstract: We show that for every e > 0 there exist δ > 0 and n 0 ∈ N such that every 3-uniform hypergraph on n ≥ n 0 vertices with the property that every k-vertex subset, where k ≥ δn, induces at least (1/4 + ɛ) edges, contains K 4 − as a subgraph, where K 4 − is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sos. The constant 1/4 is the best possible.

Integral foliated simplicial volume of aspherical manifolds

Abstract: Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.

On entropy of dynamical systems with almost specification

Abstract: We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.

The smallest singular value of random rectangular matrices with no moment assumptions on entries

Abstract: Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random matrix A = (aij) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value sn of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.

Dominant dimensions, derived equivalences and tilting modules

Abstract: The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that under a sufficient condition the infinity of dominant dimensions can be preserved by tilting, and get not only a class of derived equivalences between two algebras such that one of them is a Morita algebra in the sense of Kerner–Yamagata and the other is not, but also the first counterexample to the question whether generalized symmetric algebras are closed under derived equivalences.

On eigenvalues of random complexes

Abstract: We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model X k (n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of k-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the eigenvalues of the higher-dimensional Laplacian capture the notion of coboundary expansion—a higher-dimensional generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov; this question was raised, for instance, by Dotterrer and Kahle. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and n ∈ N, there is a k-dimensional complex Y on n vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised k-dimensional Laplacian lie in the interval $$[1 - O\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt n }}} \right. \kern- ulldelimiterspace} {\sqrt n }}} \right),\;1 + O\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt n }}} \right. \kern- ulldelimiterspace} {\sqrt n }}} \right)])$$ but whose coboundary expansion is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Y can be taken to have vanishing integer homology in dimension less than k.

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

Abstract: For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.

An approximate version of the tree packing conjecture

Abstract: We prove that for any pair of constants e > 0 and ∆ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most ∆, and with at most (n2) edges in total packs into K(1+e)n. This implies asymptotic versions of the Tree Packing Conjecture of Gy´arf´as from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

On sharp bounds for marginal densities of product measures

Abstract: We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on R n of the form f(x) = П =1 f i (x i ), where each f i is a density on R, say bounded by one, then the density of any marginal π E (f) is bounded by 2 k/2, where k is the dimension of E. The proof relies on an adaptation of Ball’s approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such f for which the cube is the extremal case.

Transitivity degrees of countable groups and acylindrical hyperbolicity

Abstract: We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here, by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.

Amenable Invariant Random Subgroups

Abstract: We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of [2]. We also consider an opposite direction, property (T), and prove a similar statement for this property.

Twins in words and long common subsequences in permutations

Abstract: A large family of words must contain two words that are similar. We investigate several problems where the measure of similarity is the length of a common subsequence.

A quantitative Oppenheim theorem for generic diagonal quadratic forms

Abstract: We establish a quantitative version of Oppenheim’s conjecture for one-parameter families of ternary indefinite quadratic forms using an analytic number-theory approach. The statements come with power gains and in some cases are essentially optimal.

Bowen entropy for actions of amenable groups

Abstract: Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Folner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.

The dynamical hierarchy for Roelcke precompact Polish groups

Abstract: We study several distinguished function algebras on a Polish group G, under the assumption that G is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of No-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urysohn sphere is Tame ∩ UC-trivial.

Spin-invariant valuations on the octonionic plane

Abstract: The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of Val2 Spin (9).

On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

Abstract: We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(ĸ)-spaces (ĸ > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.

Specialization results and ramification conditions

Abstract: Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T) with group G such that E/k is regular, we produce some specializations of E/k(T) at points t0 ∈ P1(k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior — ramified or unramified — at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T).

Moment problem in infinitely many variables

Abstract: The multivariate moment problem is investigated in the general context of the polynomial algebra R[x i | i ∈ Ω] in an arbitrary number of variables x i , i ∈ Ω. The results obtained are sharpest when the index set Ω is countable. Extensions of Haviland’s theorem [17] and Nussbaum’s theorem [34] are proved. Lasserre’s description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of R[x i | i ∈ Ω] in [27] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author in [30], [32] and [33]. Various results proved in [30], [32] and [33] are shown to continue to hold in this more general setting.

The Hurewicz dichotomy for generalized Baire spaces

Abstract: By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ww. We consider the analogous statement (which we call the Hurewicz dichotomy) for ∑11j subsets of the generalized Baire space κκ for a given uncountable cardinal κ with κ = κ<κ. We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability.

Resonances and density bounds for convex co-compact congruence subgroups of SL 2 ( ℤ )

Abstract: Let Γ be a convex co-compact subgroup of SL 2(Z), and let Γ(q) be the sequence of “congruence” subgroups of Γ. Let R q ⊂ C be the resonances of the hyperbolic Laplacian on the “congruence” surfaces Γ(q)H2. We prove two results on the density of resonances in R q as q → ∞: the first shows at least C q 3 resonances in slowly growing discs, the other one is a bound from above in boxes {δ/2 < σ = Re(s) = δ}, with |Im(s) - T| = 1, where we prove a density estimate of the type $$O\left( {{T^{\delta - {\varepsilon _1}\left( \sigma \right)}}{q^{3 - {\varepsilon _2}\left( \sigma \right)}}} \right)$$ with ej(σ) > 0 for all σ > δ/2, j = 1, 2. These two estimates highlight the role of the critical line $$\left\{ {\operatorname{Re} \left( s \right) = \frac{\delta }{2}} \right\}$$ when looking at congruence families.

Invariant convex sets in polar representations

Abstract: We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ⊂ p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.

Projections of self-similar sets with no separation condition

Abstract: We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ R d behave under linear images. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite, then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under the image of the projection. In general, with no restrictions on T we establish that H t (L o O(K)) = H t (L(K)) for every element O of the closure of T, where L is a linear map and t = dim H K. We also prove that for disjoint subsets A and B of K we have that H t (L(A) ⋂ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d,R) and the strong separation condition is satisfied, then dimH (g(K)) = min {dim H K, l} where g is a continuously differentiable map of rank l. We deduce the same result without any separation condition and we generalize a result of Eroglu by obtaining that H t (g(K)) = 0.

Coloring, sparseness and girth

Abstract: An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k - 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth.

Infinite-dimensional diagonalization and semisimplicity

Abstract: We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

Conjugacy languages in groups

Abstract: We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin

The conjugacy problem in extensions of Thompson’s group F

Abstract: We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Goncalves in [4] showing that F has property R∞, and which can be extended to show that Thompson’s group T also has property R∞.