Showing papers in "Israel Journal of Mathematics in 2019"

The trace of the canonical module

Abstract: The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen-Macaulay ring. We call a local Cohen-Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen-Macaulay K-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

Structural properties of subadditive families with applications to factorization theory

Abstract: Let H be a multiplicatively written monoid. Given k ∈ N+, we denote by $${{\mathscr U}_k}$$ the set of all l ∈ N+ such that a1 ··· ak = b1 ··· bl for some atoms (or irreducible elements) a1, … ak, b1, …, bl ∈ H. The sets $${{\mathscr U}_k}$$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large k, which is usually expressed by saying that H satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we will show that, under mild assumptions on H, not only does the Structure Theorem for Unions hold, but there also exists μ ∈ N+ such that, for every M ∈ N, the sequences $${\left({\left({{{\mathscr U}_k} - \inf \;{{\mathscr U}_k}} \right) \cap \left[\kern-0.15em\left[{0,\;M} \right]\kern-0.15em\right]} \right)_{k \ge 1}}\;\;\;\;{\rm{and}}\;\;\;\;{\left({\left({\sup \;{{\mathscr U}_k} - {{\mathscr U}_k}} \right) \cap \left[\kern-0.15em\left[{0,\;M} \right]\kern-0.15em\right]} \right)_{k \ge 1}}$$ are μ-periodic from some point on. The result applies, for instance, to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a “purely additive model” (where no explicit reference to monoids or atoms is ever made), by inquiring into the properties of what we call a subadditive family, i.e., a collection ℒ of subsets of N such that, for all L1, L2 ∈ ℒ, there is L ∈ ℒ with L1 + L2 ⊆ L.

Strongly Dependent Ordered Abelian Groups and Henselian Fields

Abstract: Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and |{p prime : [G:pG]=∞}|<∞. We apply this to show that if K is a strongly dependent field, then (K, v) is strongly dependent for any Henselian valuation v.

Periods and factors of weak model sets

Abstract: There is a renewed interest in weak model sets due to their connection to $$\mathcal{B}$$ -free systems [10], which emerged from Sarnak’s program on the M¨obius disjointness conjecture. Here we continue our recent investigation [22] of the extended hull $$\mathcal{M}^G_W$$ , a dynamical system naturally associated to a weak model set in an abelian group G with relatively compact window W. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of $$\mathcal{M}^G_W$$ and give a sufficient condition when $$\mathcal{M}^G_W$$ is an almost 1–1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system $$\mathcal{M}^G_W$$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on $$\mathcal{M}^G_W$$ , we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural G-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for $$\mathcal{B}$$ -free systems.

Envy-free cake division without assuming the players prefer nonempty pieces

Abstract: Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e., such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal- Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.

Continuity of Lyapunov exponents in the C0 topology

Abstract: We prove that the Bochi–Mane theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.

Equilateral triangles in subsets of ℝd of large Hausdorff dimension

Abstract: We prove that subsets of ℝd, d ≥ 4 of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see [3]). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open. Some interesting parallels exist between the triangle problem in Euclidean space and its counterpart in vector spaces over finite fields. We shall outline these similarities in hopes of eventually achieving a comprehensive understanding of this phenomenon in the setting of locally compact abelian groups.

Angles of Gaussian primes

Abstract: Fermat showed that every prime p = 1 mod 4 is a sum of two squares: p = a2 + b2. To any of the 8 possible representations (a, b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950’s Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.

On the Hausdorff dimension of pinned distance sets

Abstract: We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension 1 (in particular, for many x ∈ A). This verifies a strong variant of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint s = 1.

Optimal quantization for the Cantor distribution generated by infinite similutudes

Abstract: Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ , for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

Using the Steinberg algebra model to determine the center of any Leavitt path algebra

Abstract: Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.

The Ramsey-Turán problem for cliques

Abstract: An important question in extremal graph theory raised by Vera T. Sos asks to determine for a given integer t ≥ 3 and a given positive real number δ the asymptotically supremal edge density ft(δ) that an n-vertex graph can have provided it contains neither a complete graph Kt nor an independent set of size δn. Building upon recent work of Fox, Loh and Zhao [The critical window for the classical Ramsey–Turan problem, Combinatorica 35 (2015), 435–476], we prove that if δ is sufficiently small (in a sense depending on t), then $${f_t}(\delta ) = \{ \begin{array}{*{20}{c}} {\frac{{3t - 10}}{{3t - 4}} + \delta - {\delta ^2}iftiseven,} \\ {\frac{{t - 3}}{{t - 1}} + \delta iftisodd.} \end{array}$$

Separable universal Banach lattices

Abstract: We construct separable universal injective and projective lattices for the class of all separable Banach lattices.

Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

Abstract: In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth is at most two. Moreover, for the second order factorizable algebraic ODEs, all their meromorphic solutions (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

Fractional flows driven by subdifferentials in Hilbert spaces

Abstract: This paper presents an abstract theory on well-posedness for time-fractional evolution equations governed by subdifferential operators in Hilbert spaces. The proof relies on a regularization argument based on maximal monotonicity of time-fractional differential operators as well as energy estimates based on a nonlocal chain-rule formula for subdifferentials. Moreover, it will be extended to a Lipschitz perturbation problem. These abstract results will be also applied to time-fractional nonlinear PDEs such as time-fractional porous medium, fast diffusion, p-Laplace parabolic, Allen-Cahn equations.

Moduli of regularity and rates of convergence for Fejér monotone sequences

Abstract: In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the linear convergence and finite termination, for classes of Fejer monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, Holder regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic projection method. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent.

The separable quotient problem for topological groups

Abstract: The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces. For a topological group G there are four natural analogous problems: Does G have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. Positive answers to all four questions are proved for groups G which belong to the important classes of (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. However, a surprising example of an uncountable precompact group G is produced which has no non-trivial separable quotient group other than the trivial group. Indeed Gτ has the same property, for every cardinal number τ ≥ 1.

Well-posedness for degenerate third order equations with delay and applications to inverse problems

Abstract: In this paper, we study well-posedness for the following third-order in time equation with delay $$\left( {0.1} \right)\;\alpha \left( {Mu'} \right)''\left( t \right) + \left( {Nu'} \right)'\left( t \right) = \beta Au\left( t \right) + \gamma Bu{'(t)} + Gu{'_t}+F{u_t} + f\left( t \right),\;t \in \left[ {0,2\pi } \right]$$ where α, β, γ are real numbers, A and B are linear operators defined on a Banach space X with domains D(A) and D(B) such that $$D(A)\cap{D(B)}\subset{D(M)}\cap{D(N)};$$ u(t)is the state function taking values in X and ut: (−∞, 0] → X defined as ut(θ) = u(t+θ) for θ < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue–Bochner spaces $$L^p(\mathbb{T}, X)$$ , periodic Besov spaces $$B^s_{p,q}(\mathbb{T}, X)$$ and periodic Triebel–Lizorkin spaces $$F^s_{p,q}(\mathbb{T}, X)$$ . A novel application to an inverse problem is given.

The back-and-forth method and computability without delay

Abstract: We study the algorithmic content of back-and-forth proofs for graphs and homogeneous structures from the perspective of Turing computations in which unbounded search is forbidden. Quite unexpectedly, we discover subtle differences between the back-and-forth proofs for the random graph, the dense linear order of the rationals, and the universal countable abelian p-group. We also prove the primitive recursive analog of the Cantor–Bernstein theorem for graphs which says that if there is a “back” isomorphism and the “forth” isomorphism, then there is a “back-and-forth” isomorphism. We also show that the fully primitive recursive degrees (to be defined) of the dense linear order of the rationals are downwards dense.

Completion and torsion over commutative DG rings

Abstract: Let CDGcont be the category whose objects are pairs ( $$(A, \bar{\mathfrak{a}})$$ ), where A is a commutative DG-algebra and $$\bar{\mathfrak{a}} \subseteq {{\rm{H}}^0}\left( A \right)$$ is a finitely generated ideal, and whose morphisms $$f:\left( {A,\bar{\mathfrak{a}}} \right) \to \left( {B,\bar{\mathfrak{b}}} \right)$$ are morphisms of DG-algebras A → B, such that $$\left( {{{\rm{H}}^0}\left( f \right)\left( {\bar{\mathfrak{a}}} \right)} \right) \subseteq \bar{\mathfrak{b}}$$ . Letting Ho(CDGcont) be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor LΛ: Ho(CDGcont) → Ho(CDGcont) which takes a pair ( $$\left( {A,\bar{\mathfrak{a}}} \right)$$ ) into its non-abelian derived $$\bar{\mathfrak{a}}$$ -adic completion. We show that this operation has, in a derived sense, the usual properties of adic completion of commutative rings, and that if A = H0(A) is an ordinary noetherian ring, this operation coincides with ordinary adic completion. As an application, following a question of Buchweitz and Flenner, we show that if $$\mathbb{k}$$ is a commutative ring, and A is a commutative $$\mathbb{k}$$ -algebra which is $$\mathfrak{a}$$ -adically complete with respect to a finitely generated ideal $$\mathfrak{a} \subseteq A$$ , then the derived Hochschild cohomology modules $$\mathrm{Ext}_{A\otimes_{\mathbb{k}}^{\mathrm{L}}A}^{n}(A,A)$$ and the derived complete Hochschild cohomology modules $$\mathrm{Ext}_{A\hat{\otimes}_{\mathbb{k}}^{\mathrm{L}}A}^{n}(A,A)$$ coincide, without assuming any finiteness or noetherian conditions on $$\mathbb{k}$$ , A or on the map $$\mathbb{k}\rightarrow A$$ .

Hypergraph cuts above the average

Abstract: An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $$m{\rm/}2 + {\rm\Omega)}(\sqrt m)$$ and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $${\rm\Omega}(\sqrt m)$$ larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m5/9) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

Hausdorff dimensions in p-adic analytic groups

Abstract: Let G be a finitely generated pro-p group, equipped with the p-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of G. Conversely, it is a long-standing open question whether the finiteness of the Hausdorff spectrum implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for odd primes p, that every countably based pro-p group G with an open subgroup mapping onto 2 copies of the p-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.

Stability and sparsity in sets of natural numbers

Abstract: Given a set A ⊆ ℕ, we consider the relationship between stability of the structure (ℤ, + , 0,A) and sparsity of the set A. We first show that a strong enough sparsity assumption on A yields stability of (ℤ, +, 0, A). Specifically, if there is a function f: A → ℝ+ such that supa∈A |a − f(a)| < ∞ and { $$\frac{s}{t}:s,t \in f(A) $$ , t ≤ s} is closed and discrete, then (ℤ, +, 0, A) is superstable (of U-rank ω if A is infinite). Such sets include examples considered by Palacin and Sklinos [19] and Poizat [23], many classical linear recurrence sequences (e.g., the Fibonaccci numbers), and any set in which the limit of ratios of consecutive elements diverges. Finally, we consider sparsity conclusions on sets A ⊆ N, which follow from model theoretic assumptions on (ℤ, +, 0, A). We use a result of Erdős, Nathanson and Sarkozy [8] to show that if (ℤ, +, 0, A) does not define the ordering on ℤ, then the lower asymptotic density of any finitary sumset of A is zero. Finally, in a theorem communicated to us by Goldbring, we use a result of Jin [11] to show that if (ℤ, +, 0,A) is stable, then the upper Banach density of any finitary sumset of A is zero.

Mean dimension of full shifts

Abstract: Let K be a finite-dimensional compact metric space and Kℤ the full shift on the alphabet K. We prove that its mean dimension is given by dimK or dimK−1 depending on the “type” of K. We propose a problem which seems interesting from the view point of infinite-dimensional topology.

Flexibility of entropies for surfaces of negative curvature

Abstract: We consider a smooth closed surface M of fixed genus ⩾ 2 with a Rie-mannian metric g of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for g is greater than or equal to the topological entropy for the metric of constant negative curvature on M with the same total area which is greater than or equal to the metric entropy with respect to the Liouville measure of geodesic flow for g. Equality holds only in the case of constant negative curvature. We prove that those are the only restrictions on the values of topological and metric entropies for metrics of negative curvature.

Representations of Lie algebras of vector fields on affine varieties

Abstract: For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

Generators of semigroups on Banach spaces inducing holomorphic semiflows

Abstract: Let A be the generator of a C0-semigroup T on a Banach space of analytic functions on the open unit disc. If T consists of composition operators, then there exists a holomorphic function G: $$\mathbb{D}$$ → ℂ such that Af = Gf′ with maximal domain. The aim of the paper is the study of the reciprocal implication.

The pressure function for infinite equilibrium measures

Abstract: Assume that (X, f) is a dynamical system and φ: X → [−∞, ∞) is a potential such that the f-invariant measure μφ equivalent to the φ-conformal measure is infinite, but that there is an inducing scheme F = fτ with a finite measure $$\mu_\phi^-$$ and polynomial tails $$\mu_\phi^-$$ (τ ≥ n) = O(n−β), β ∈ (0, 1). We give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system as $$P(\phi + s\psi ) = (CP{(\overline {\phi + s\psi )} )^{1/\beta }}(1 + o(1)),$$ together with estimates for the o(1)-error term. This extends results from Sarig [S06] to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes [L93] for the Pomeau-Manneville map with potential φt = −tlogf′, as well as on the results by Bruin and Todd [BTo09, BTo12] on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures μφ+sψ as s → 0 are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.

Small sets in dense pairs

Abstract: Let $$\widetilde{\cal M} = \langle {\cal M},P\rangle $$ be an expansion of an o-minimal structure $$\mathcal M$$ by a dense set P ⊆ M, such that three tameness conditions hold. We prove that the induced structure on P by $$\mathcal M$$ eliminates imaginaries. As a corollary, we obtain that every small set X definable in $$\widetilde{\cal M}$$ can be definably embedded into some Pl, uniformly in parameters, settling a question from [8]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of ℳ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [13]. The above results are in contrast to recent literature, as it is known in general that $$\widetilde{\cal M}$$ does not eliminate imaginaries, and neither it nor the induced structure on P admits definable Skolem functions.

Maximal function inequalities and a theorem of Birch

Abstract: In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic hypersurfaces. Let p be a homogenous polynomial in n variables with integer coefficients of degree d > 1. The maximal functions we consider are defined by $${A_*}f(y) = \begin{array}{*{20}{c}} {\sup } \\ {N \geq 1} \end{array}|\frac{1}{{r(N)}}\sum\limits_{p(x) = 0;x \in {{[N]}^n}} {f(y - x)|} $$ for functions f : ℤn → ℂ, where [N] = {−N,−N + 1, …, N} and r(N) represents the number of integral points on the surface defined by p(x) = 0 inside the n-cube [N]n. It is shown here that the operators A* are bounded on lp in the optimal range p > 1 under certain regularity assumptions on the polynomial p.