Showing papers in "Israel Journal of Mathematics in 2020"

A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity

Abstract: We utilize a new definition for the fractional delta operator and prove that it is equivalent by translation to the more commonly used operator. By means of the convolution operation we demonstrate that this new operator is strongly connected to the positivity, monotonicity, and convexity of the functions on which it operates. We also analyze the case of compositions of discrete fractional operators. Finally, since the operator we study here is translationally related to the more commonly used discrete fractional operators, we are able to establish many new results for all types of discrete fractional differences, and we explicitly demonstrate that our results improve all known existing results in the literature.

A family of q -hypergeometric congruences modulo the fourth power of a cyclotomic polynomial

Abstract: We prove a two-parameter family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews’ multiseries extension of the Watson transformation, and a Karlsson—Minton-type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of q-congruences is then used to prove two conjectures posed earlier by the authors.

On hierarchical hyperbolicity of cubical groups

Abstract: Let χ be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that χ has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that G is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that χ cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.

Nearly orthogonal vectors and small antipodal spherical codes

Abstract: How can d+k vectors in ℝd be arranged so that they are as close to orthogonal as possible? In particular, define θ(d, k) := minX maxx≠y∈X |〈x, y〉 | where the minimum is taken over all collections of d + k unit vectors X ⊆ ℝd. In this paper, we focus on the case here k is fixed and d → ∞. In establishing bounds on θ(d, k), we find an intimate connection to the existence of systems of $$\left(\begin{array}{c}k+1\\ 2\end{array}\right)$$ equiangular lines in ℝk. Using this connection, we are able to pin down θ(d, k) whenever k ∈ {1, 2, 3, 7, 23} and establish asymptotics for general k. The main tool is an upper bound on $$\mathbb{E}_{x,y\sim\mu}|\langle{x,y}\rangle|$$ whenever μ is an isotropic probability mass on ℝk, which may be of independent interest. Our results translate naturally to the analogous question in ℂd. In this case, the question relates to the existence of systems of k2 equiangular lines in ℂk, also known as SIC-POVM in physics literature.

GCD sums and sum-product estimates

Abstract: In this note we prove a new estimate on so-called GCD sums (also called Gal sums), which, for certain coefficients, improves significantly over the general bound due to de la Breteche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.

Domination, almost additivity and thermodynamical formalism for planar matrix cocycles

Abstract: In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

Finite groups, 2-generation and the uniform domination number

Abstract: Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1,…,xk, there exists y G ∈ G such that G = 〈xi,y〉 for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) ⩾ 2 for every non-abelian finite simple group G. For any group with u(G) ⩾ 1, we define the uniform domination number γu(G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈x, y〉 (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu(G) for all simple groups G. In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2-generation of soluble and symmetric groups, and we present several open problems.

On Erdős-Ginzburg-ZIV inverse theorems for dihedral and dicyclic groups

Abstract: Let G be a finite group and exp(G) = lcm{ord(g) ∣ g ∈ G}. A finite unordered sequence of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. We denote by s(G)(or E(G) respectively) the smallest integer l such that every sequence of length at least l has a product-one subsequence of length exp(G)(or ∣G∣ respectively). In this paper, we provide the exact values of s(G) and E(G) for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length s(G) — 1 (or E(G) — 1 respectively) having no product-one subsequence of length exp(G)(or ∣G∣ respectively).

Popular products and continued fractions

Abstract: We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, considering Zaremba’s set modulo p, that is the set of all a such that $${a \over p} = [{a_1}, \ldots ,{a_s}]$$ has bounded partial quotients, aj ⩽ M, we obtain a sharp upper bound for the cardinality of this set.

Local rigidity of Lyapunov spectrum for toral automorphisms

Abstract: We study the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L4 is irreducible over ℚ. We consider a volume-preserving C1-small perturbation f of L. We show that if Lyapunov exponents of f with respect to the volume are the same as Lyapunov exponents of L, then f is C1+Holder conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.

Metric properties of the product of consecutive partial quotients in continued fractions

Abstract: In the one-dimensional Diophantine approximation, by using the continued fractions, Khintchine’s theorem and Jarnik’s theorem are concerned with the growth of the large partial quotients, while the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. This paper aims to establish a complete characterization on the metric properties of the product of the partial quotients, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result. More precisely, for any x ∈ [0, 1), let x =[a1, a2, …] beits continued fraction expansion. The size of the following set, in the sense of Lebesgue measure and Hausdorff dimension, Em(ϕ):= {x ∈ [0, 1): an (x) ⋯ an+m−1 (x) ≥ ϕ(n) for infinitely many n ∈ ℕ}, are given completely, where m ≥ 1 is an integer and ϕ: ℕ → ℝ+ is a positive function.

Slice Dirac operator over octonions

Abstract: The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

Lipschitz free p-spaces for 0 < p < 1

Abstract: F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM2014-53009-P for Analisis Vectorial, Multilineal y Aplicaciones, and MTM2016-76808-P for Operators, lattices, and structure of Banach spaces as well as the Spanish Ministry for Science and Innovation under Grant PID2019-1077701GB-I00. J. L. Ansorena acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2014-53009-P for Analisis Vectorial, Multilineal y Aplicaciones. M. Cuth has been supported by Charles University Research program No. UNCE/SCI/023 and by the Research grant GACR 17-04197Y. M. Doucha was supported by the GACR project 16-34860L and RVO: 67985840.

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

Abstract: We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link thedoublecosets with Cech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass’s finiteness conjecture in K-theory.

Nonlinear descent on moduli of local systems

Abstract: We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.

Proof of the Kalai-Meshulam conjecture

Abstract: Let G be a graph, and let fG be the sum of (−1)∣A∣, over all stable sets A. If G is a cycle with length divisible by three, then fG = ±2. Motivated by topological considerations, G. Kalai and R. Meshulam [8] made the conjecture that, if no induced cycle of a graph G has length divisible by three, then ∣fG∣ ≤ 1. We prove this conjecture.

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

Abstract: Under various conditions on the data we analyze how the appearance of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form $$- {\rm{div}}\;a\left( {x,Du} \right) + b\left( {x,u} \right) = \mu$$ with data μ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function whose speed of growth modulates the regularization of the solutions.

Spatio-temporal Poisson processes for visits to small sets

Abstract: For many measure preserving dynamical systems (Ω,T, μ) the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times n of visits to a set A, but also the position Tn (x)in A of the orbit, in the limit where μ(A) → 0. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.

Intersection of unit balls in classical matrix ensembles

Abstract: We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschlager for classical lp-balls [Schechtman and Schmuckenschlager, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.

The number of 4-colorings of the Hamming cube

Abstract: Let Qd be the d-dimensional hypercube and N = 2d. We prove that the number of (proper) 4-colorings of Qd is asymptotically 6e2N, as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980’s.

Nearly-linear monotone paths in edge-ordered graphs

Abstract: How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvatal and Komlos in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1−o(1).

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

Abstract: In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e., a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied by the first two authors and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Our approach also works to show undecidability of the freeness problem for automaton semigroups, which negatively solves an open problem by Grigorchuk, Nekrashevych and Sushchansky. In fact, we show undecidability of a strengthened version where the input automaton is complete and invertible. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable.

On Riemann surfaces of genus $g$ with $4g-4$ automorphisms

Abstract: In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order, namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g–4. Under the assumption that g–1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

Coherent actions by homeomorphisms on the real line or an interval

Abstract: We study actions of groups by orientation preserving homeomorphisms on R (or an interval) that are minimal, have solvable germs at ±∞ and contain a pair of elements of a certain dynamical type. We call such actions coherent. We establish that such an action is rigid, i.e., any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non-elementary amenable, but satisfies that every proper quotient is solvable. The structure theory we develop allows us to prove a plethora of non-embeddability statements concerning groups of piecewise linear and piecewise projective homeomorphisms. For instance, we demonstrate that any coherent group action G < Horneo+ (R) that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson’s group F. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. We also establish that the Brown-Stein-Thompson groups F(2, pi,…,pn) for n ≥ 1 and p1,…,pn distinct odd primes, do not embed into Thompson’s group F. This answers a question recently raised by C. Bleak, M. Brin and J. Moore.

Counting problems in graph products and relatively hyperbolic groups

Abstract: We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e., they have full density with respect to counting in balls for the word metric in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups.

Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines

Abstract: The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $${\mathcal F}\left( \lambda \right)$$ be the family of connected graphs of spectral radius ≤ λ. We show that $${\mathcal F}\left( \lambda \right)$$ can be defined by a finite set of forbidden subgraphs if and only if $$\lambda > \lambda *: = \sqrt {2 + \sqrt 5 } \approx 2.058$$ and λ ∉ {α2, α3, …}, where $${\alpha _m} = \beta _m^{1/2} + \beta _m^{ - 1/2}$$ and βm is the largest root of xm+1 = 1+ x + … + xm−1. The study of forbidden subgraphs characterization for $${\mathcal F}\left( \lambda \right)$$ is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space ℝn family of lines through the origin such that the angle between any pair of them is the same. Denote by Nα(n) the maximum number of equiangular lines in ℝn with angle arccos α. We establish the asymptotic formula Nα(n) = cαn + Oα(1) for every $${N_\alpha }\left( n \right) = {c_\alpha }n + {O_\alpha }\left( 1 \right)$$. In particular, $$\alpha \ge {1 \over {1 + 2\lambda *}}$$. Besides we show that $${N_{1/3}}\left( n \right) = 2n + O\left( 1 \right)\quad {\rm{and}}\quad {N_{1/5}}\left( n \right),\,{N_{1/(1 + 2\sqrt 2 )}}(n) = {3 \over 2}n\, + O\left( 1 \right).$$, which improves a recent result of Balla, Draxler, Keevash and Sudakov.

Ergodicity and central limit theorem for random interval homeomorphisms

Abstract: The central limit theorem for Markov chains generated by iterated function systems consisting of orientation-preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.

Structure theorems in tame expansions of o-minimal structures by a dense set

Abstract: We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$ -structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such $$\widetilde{\cal M}$$ , as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $${\cal L}$$ -definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an $${\cal L}$$ -definable map.

Ranks of elliptic curves over $$\mathbb{Z}_p^2$$ℤp2-extensions

Abstract: Let E be an elliptic curve with good reduction at a fixed odd prime p and K an imaginary quadratic field where p splits. We give a growth estimate for the Mordell-Weil rank of E over finite extensions inside the $$\mathbb{Z}_p^2$$-extension of K.

On the question “Can one hear the shape of a group?” and a Hulanicki type theorem for graphs

Abstract: We study the question of whether or not it is possible to determine a finitely generated group G up to some notion of equivalence from the spectrum sp(G) of G. We show that the answer is “No” in a strong sense. As a first example we present the collection of amenable 4-generated groups Gω, ω ∈ {0, 1, 2}ℕ, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $${\rm{sp}}(G_\omega)=[-\frac{1}{2},0]\cup[\frac{1}{2},1]$$ . Moreover, for each of these groups Gω there is a continuum of covering groups G with the same spectrum. As a second example we construct a continuum of 2-generated torsion-free step-3 solvable groups with the spectrum [-1, 1]. In addition, in relation to the above results, we prove a version of the Hulanicki Theorem about inclusion of spectra for covering graphs.