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Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics

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 $x_1, \ldots, x_k$, there exists $y \in G$ such that $G = \langle x_i, y\rangle$ 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 $u(G) \geqslant 2$ for every non-abelian finite simple group $G$. For any group with $u(G) \geqslant 1$, we define the uniform domination number $\gamma_u(G)$ of $G$ to be the minimal size of a subset $S$ of conjugate elements such that for each nontrivial $x \in G$ there exists $y \in S$ with $G = \langle x, y \rangle$ (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 $\gamma_u(G)$ 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 $\gamma_u(G)=2$, 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., Comment: 60 pages; to appear in Israel Journal of Mathematics

Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper., Comment: The section concerning stabilization of model categories was separated into an independent paper, appearing now as arXiv:1802.08031

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.

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices., Comment: 23 pages, revised following referee's comments. Dedicated to Aner Shalev on his 60th birthday. This paper is related to our previous work arXiv:0905.1841 with which it shares some preliminaries

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring., Comment: 46 pages

For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that the distribution of the pair correlations of the fractional parts of $\{ \langle \alpha a(x) \rangle \}_{x \geq 1}$ is asymptotically Poissonian for almost all $\alpha$ if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of $\alpha$ such that $\{\langle \alpha x^d \rangle\}$ fails to have Poissonian pair correlation is at most $\frac{d+2}{d+3} < 1$. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $\frac{2}{d+1}$. An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved., Comment: 22 pages. Version 2 contains an appendix by Jean Bourgain. Version 3 with corrections suggested by the referee. The paper will be published in the Israel Journal of Mathematics

This paper explores the consistency strength of The Proper Forcing Axiom ($\textsf{PFA}$) and the theory (T) which involves a variation of the Viale-Wei$\ss$ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper implies that the theory "$\sf{AD}$$_\mathbb{R} + \Theta$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".

It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra $A=\kk[z;\lambda,\eta,\varphi(z)]$ is homologically smooth, then the polynomial $\varphi(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when $\kk$ is of characteristic zero., Comment: Final version, 36 pages

This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context., 33 pages, 1 figure

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type., 27 pages, no figures, submitted, Israel J. Math. It turns out that there is one specific family of algebraic group - namely Sp(4) in characteristic 3 - which our methods are unable to resolve, and we have adjusted the paper to exclude this case from the results

In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper., 13 pages

We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations. Let $X$ be an equidimensional projective variety of dimension $d$. For a given $k \leq d + 1$, we are interested in the study of the variety of $k$-secants. The classical trisecant lemma just considers the case where $k = 3$ while elsewhere the case $k = d + 2$ is considered. Secants of order from $4$ to $d + 1$ provide service for our main result. In this paper, we prove that if the variety of $k$-secants ($k \leq d + 1$) satisfies the three following conditions: (i) trough every point in $X$, passes at least one $k$-secant, (ii) the variety of $k$-secant satisfies a strong connectivity property that we defined in the sequel, (iii) every $k$-secant is also a ($k+1$)-secant, then the variety $X$ can be embedded into $P^{d+1}$. The new assumption, introduced here, that we called strong connectivity is essential because a naive generalization that does not incorporate this assumption fails as we show in some example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption., Comment: arXiv admin note: text overlap with arXiv:0712.3878

In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series., Comment: to appear in Israel Journal of Mathematics

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method., 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Math

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.

$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lov\'asz to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have $\varpi_1^k(\thom(K_m,G))\neq 0$, then $\chi(G)\geq k+m$; here $Z_2$-action is induced by the swapping of two vertices in $K_m$, and $\varpi_1$ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of $Hom(G,H)$ induces a homotopy equivalence. It then follows that $Hom(F,K_n)$ is homotopy equivalent to a direct product of $(n-2)$-dimensional spheres, while $Hom(\bar{F},K_n)$ is homotopy equivalent to a wedge of spheres, where $F$ is an arbitrary forest and $\bar{F}$ is its complement., Comment: This is the first part of the series of papers containing the complete proofs of the results announced in "Topological obstructions to graph colorings". This is the final version which is to appear in Israel J. Math., it has an updated list of references and new remarks on latest developments

The striking results on noncrossed products were their existence (Amitsur) and the determination of Q(t) and Q((t)) as their smallest possible centres (Brussel). This paper gives the first fully explicit noncrossed product example over Q((t)). As a consequence, the use of deep number theoretic theorems (local-global principles such as the Hasse norm theorem and density theorems) in order to prove existence is eliminated. Instead, the example can be verified by direct calculations. The noncrossed product proof is short and elementary., Comment: 4 pages (A4), updated version of published paper, the sign error in the definition of the element \pi has been corrected

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex.

Consider the generalized absolute value function defined by \[ a(t) = \vert t \vert t^{n-1}, \qquad t \in \mathbb{R}, n \in \mathbb{N}_{\geq 1}. \] Further, consider the $n$-th order divided difference function $a^{[n]}: \mathbb{R}^{n+1} \rightarrow \mathbb{C}$ and let $1 < p_1, \ldots, p_n < \infty$ be such that $\sum_{l=1}^n p_l^{-1} = 1$. Let $\mathcal{S}_{p_l}$ denote the Schatten-von Neumann ideals and let $\mathcal{S}_{1,\infty}$ denote the weak trace class ideal. We show that for any $(n+1)$-tuple ${\bf A}$ of bounded self-adjoint operators the multiple operator integral $T_{a^{[n]}}^{\bf A}$ maps $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1, \infty}$ boundedly with uniform bound in ${\bf A}$. The same is true for the class of $C^{n+1}$-functions that outside the interval $[-1, 1]$ equal $a$. In [CLPST16] it was proved that for a function $f$ in this class such boundedness of $T^{ {\bf A} }_{f^{[n]}}$ from $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1}$ may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences., to appear in Israel Journal of Mathematics

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present paper to Seifert fibrations and we focused only to topological and number theoretic aspects related to Froyshov invariants, Latex 2.09, 57 pages, 6 figures

An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $\eta \geq 0$, $\Pr(s_n(M_n) \leq \eta) \lesssim n^{C}\eta + \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a `truly combinatorial' setting., Comment: 28 pages; comments welcome!

A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona., Comment: We split our original paper (arXiv:1807.00793v2) into two parts. The first part can be found in arXiv:1807.00793. This is the second part, which consists of 18 pages, including a two-page appendix

Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of vertices of the subgraphs gives rise to the definition of the normalized cyclomatic quotient $\Xi (G^{\alpha})$. The asymptotic behavior of this quotient is similar to the asymptotic behavior of the quotient of the cardinality of the boundary of the subgraph by the cardinality of the subgraph. Using Følner's criterion for amenability one gets that $\Xi (G^{\alpha})$ vanishes for infinite groups if and only if they are amenable. When $G$ is finite then $\Xi (G^{\alpha})=1/|G|$, where $|G|$'> is the cardinality of $G$, and when $G$ is non-amenable then $1-n\leq\Xi (G^{\alpha})\le 0$, with $\Xi (G^{\alpha})=1-n$ if and only if $G$ is free of rank $n$. Thus we see that on special cases $\Xi (G^{\alpha})$ takes the values of the Euler characteristic of $G$. Most of the paper is concerned with formulae for the value of $\Xi (G^{\alpha})$ with respect to that of subgroups and factor groups, and with respect to the decomposition of the group into direct product and free product. Some of the formulae and bounds we get for $\Xi (G^{\alpha})$ are similar to those given for the spectral radius of symmetric random walks on the graph of $G^{\alpha}$, but this is not always the case. In the last section of the paper we define and touch very briefly the balanced cyclomatic quotient, which is defined on concentric balls in the graph and is related to the growth of $G$., Comment: LaTex, 23 pages, no figures

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section., Comment: 38 pages, comments are welcome - minor typos were corrected in the second version

In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th unknown in $i$-th equation is less than the number unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianisation into an algebraic group $G$ is at least $\dim G$., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: An open question added, misprints corrected. V.3: misprints corrected

In this paper, we determine the topological entropy $h(\phi)$ of a continuous endomorphism $\phi$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\phi)$, which is characterized by the so called variational principle. We prove that \[ h\left(\phi\right) = h\left(\phi|_{T(G_\phi)}\right) \] where $G_\phi$ is the maximal connected subgroup of $G$ such that $\phi(G_\phi) = G_\phi$, and $T(G_\phi)$ is the maximal torus in the center of $G_\phi$. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.

In this paper we prove that the Ziegler spectra of all serial rings are sober. We then use this proof to give a general framework for computing and understanding Ziegler spectra of uniserial rings up to topological indistinguishability. Finally, we illustrate this technique by computing the Ziegler spectra of all rank one uniserial domains up to topological indistinguishability., Comment: 30 pages

Let $A$ be the generator of a $C_0$-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 $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication., 14 pages, Accepted, Israel Journal of Mathematics 2018

In this paper we prove a quantum generalization of Regev's theorems in (Israel. J. Math. 195 (2013), 31--35) by applying the Schur-Weyl duality between the quantum superalgebra and Iwahori-Hecke algebra. We also present an alternative proof of the quantized generalizations using the skew character theory of Iwahori-Hecke algebras., Final version: added a new section 5 to contain the algebraic proof of arXiv:1710.02778

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over $\emptyset$ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $\mathfrak{C}$ be a monster model of a countable theory, $p \in S(\emptyset)$, and $E$ be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on $p(\mathfrak{C})$. Then, exactly one of the following holds: (1) $E$ is relatively definable (on $p(\mathfrak{C})$), smooth, and has finitely many classes, (2) $E$ is not relatively definable, but it is type-definable, smooth, and has $2^{\aleph_0}$ classes, (3) $E$ is not type definable and not smooth, and has $2^{\aleph_0}$ classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups., 57 pages

In this paper, we show the existence of universal inequalities for the $h^*$-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the $h^*$-polynomial which are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients $h^*_1$ and $h^*_2$ of the $h^*$-vector $(h^*_0,h^*_1,\ldots,h^*_d)$ of a lattice polytope of any degree satisfy Scott's inequality if $h^*_3=0$., Comment: 9 pages, 1 figure

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $\mu$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(\mu)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption., Comment: final version, new co-author, to appear in: Israel Journal of Mathematics

In this paper, we prove that for any totally real field $F$, weight $k$, and nebentypus character $\chi$, the proportion of Hilbert cusp forms over $F$ of weight $k$ and character $\chi$ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible $GL_2$ representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for $GL_2$; and third, a Plancherel equidsitribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin-Templier and mirrors work of Finis-Lapid-Mueller by introducing an explicit bound for certain families of orbital integrals., Comment: 41 pages

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured $K_n$ formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties., 9 pages, 1 figure

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kuhn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r (1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K n is $${n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}$$ .

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation with negatively curved leaves. On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when this foliated geodesic flow preserves a Gibbs su-state, i.e. a measure with Lebesgue disintegration both in the stable and unstable horospheres, then it has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. On the other hand we study in detail the projections of Gibbs u-states along the unit spheres tangent to the foliation. We show that they have Lebesgue disintegration in the leaves and that the local densities possess an integral representation analogue to the Poisson representation of harmonic functions., Comment: 50 pages, 13 figures, to appear in Israel Journal of Mathematics

We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper., Comment: 9 pages

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.

In this sequel to Part-I, we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes' rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part-I in this series., Comment: 26 pages

In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra., 25 pages

We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by Kaimanovich-Vershik and Karlsson-Ledrappier. We then proceed to prove that the product of the Poisson boundary of any countable measured group $(G,\mu)$ with any ergodic $(G,\check{\mu})$-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of $(G,\mu)$ when $\mu$ is symmetric. Finally, we characterize the failure of weak-mixing for an ergodic $(G,\mu)$-space as the existence of a non-trivial measure-preserving isometric factor., Comment: 8 pages, no figures. Substantial overlap with the (longer) paper "Five remarks about random walks on groups", http://arxiv.org/abs/1406.0763

In this article we use the adiabatic method to prove the gluing formula of real analytic torsion forms for a flat vector bundle on a smooth fibration under the assumption that the fiberwise twisted cohomology groups associated to the fibration of the cutting hypersurface are vanished. In this paper we assume that the metrics have product structures near the cutting hypersurface.

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 the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology., Comment: We added an Appendix by Phillip Wesolek

The present paper is devoted to the study of classes of mappings with non--bounded characteristics of quasiconformality. It is proved that the normal families of mappings distorting the families of mappings in ${\Bbb R}^n$ by special way, have the logarithmic order of growth in the neighborhood of every point. There are proved some sufficient conditions of normality of such mappings $f:D\rightarrow \bar{{\Bbb R}^n},$ $n\ge 2,$ omitting the values of some set $E_f$ with some constraints of the type $c(E_f)\ge \delta,$ $\delta\in {\Bbb R},$ where $c(\cdot)$ is the special set function., Comment: 16 pages

The classical Waring problem deals with expressing every natural number as a sum of g(k) k th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w ≠ 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed word w ≠ 1 and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)2 = G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over ℝ or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover., 18 pages

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for $\lambda (G)$ appear in the study of various problems on finite, residually finite, and profinite groups. We prove that $\lambda (G)$ is bounded in terms of the maximum $2$-length of soluble subgroups of $G$, and that $\lambda (G)$ is bounded by the maximum Fitting height of soluble subgroups. For an odd prime $p$, the non-$p$-soluble length $\lambda _p(G)$ is introduced, and it is proved that $\lambda _p(G)$ does not exceed the maximum $p$-length of $p$-soluble subgroups. We conjecture that for a given prime $p$ and a given proper group variety ${\frak V}$ the non-$p$-soluble length $\lambda _p(G)$ of finite groups $G$ whose Sylow $p$-subgroups belong to ${\frak V}$ is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent., Comment: some definitions amended and some misprints corrected