General discrepancy estimates III: The Erdös-Turán-Koksma inequality for the Haar function system

The classical Erdös-Turán-Koksma inequality gives us an upper bound for the discrepancy of a sequence in thes-dimensional unit cube in terms of exponential sums, more precisely, in terms of the trigonometric function system.In this paper, we shall prove the inequality of Erdös-Turán-Koksma for the extreme and the star discrepancy, for generalized Haar function systems. Further, we shall show the existence of the inequality of Erdös-Turán-Koksma for the isotropic discrepancy, for generalized Haar and Walsh function systems.Research supported by the Austrian Science Foundation, project no. P9285/TEC.     

A generalization of the Erdös-Ko-Rado theorem

In this note, we investigate some properties of local Kneser graphs defined in [János Körner, Concetta Pilotto, Gábor Simonyi, Local chromatic number and sperner capacity, J. Combin. Theory Ser. B 95 (1) (2005) 101-117]. In this regard, as a generalization of the Erdös-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we provide an upper bound for their chromatic number.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

Erdös-Turán theorems on a system of Jordan curves and arcs

Erdös and Turán established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [–1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadec's theorem) are given.Communicated by Dieter Gaier.     

Invariant measure and Lyapunov exponents for birational maps of P2

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261-303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417-424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts.     

Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.     

Constantes d’Erdös–Turán

L’inégalité d’Erdös-Turán mesure l’écart à l’équirépartition d’une suite quelconque du tore en fonction d’un paramètre arbitraire et de deux constantes absolues, c₁ et c₂. Nous montrons que c₁≥ 1 et c₂≥ 2/π, et nous fournissons un ensemble de couples admissibles (c₁;c₂) numériquement proches de l’optimum hypothétique (1;2/π), notamment (1;0,653) et (1,1435;2/π).The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, c₁ and c₂. We show that c₁≥ 1 and c₂≥ 2/π, and we provide a set of admissible pairs (c₁;c₂) that are numerically close to the hypothetical optimum (1;2/π), including (1;0.653) and (1.1435;2/π).À Jean-Louis Nicolas, avec toute notre amitié2000 Mathematics Subject Classification: Primary—11K38, 11K06; Secondary—11L03, 42A05     

A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the same type lemma, which was proved by Bárány and Valtr.     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique

We prove that admissible functions for Fubini-Study metric on the complex projective space PmC of complex dimension m, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of PmC. A similar lower bound holds on some projective manifolds. This gives an optimal constant in a Hörmander type inequality on these manifolds, which allows us, using Tian's invariant, to establish the existence of Einstein-Kähler metrics on them.     

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83-109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88-112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261-280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359-392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.     

On Equitorsion Holomorphically Projective Mappings of Generalized Kählerian Spaces

In this paper we investigate holomorphically projective mappings of generalized Kählerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kählerian spaces we obtain five invariant geometric objects for these mappings.     

A note on badly approximabe sets in projective space

Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result also holds for ‘badly approximable’ points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine’s theorem for convergence relating to ‘intrinsic’ approximation of points in these compact sets.     

Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

Geometries with dual affine planes and symplectic quadrics

Farmer and Hale [3] prove that every copolar space fully embedded in a finite projective space PG(n, q), with q>, is the copolar space arising from a symplectic polarity. We show that this result is still valid in arbitrary projective spaces; this provides a different and shorter proof of [3] in the finite case.     

On entropy of dynamical systems with almost specification

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.     

On the discrepancy of uniformly distributed roots of quadratic congruences

In this paper we consider the distribution of fractional parts {ν/p}, where p is a prime less than or equal to x and ν is the root in Z/pZ of a quadratic polynomial with negative discriminant. This set is known to be uniformly distributed as x→∞. Here we apply the Erd?s-Turán inequality to obtain an estimate for the discrepancy.     

Varying Jacobi–Krall orthogonal polynomials: local asymptotic behaviour and zeros

Krall orthogonal polynomials are well known and they constitute a generalization of classical orthogonal polynomials obtained by addition of positive masses located at some points on the real line. In this contribution we consider two families of Krall polynomials already known in the literature, but now the corresponding absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the point 1 in the Jacobi case and at the end points of the interval of orthogonality in the Gegenbauer case. We analyze the asymptotic behaviour of these varying Krall orthogonal polynomials in the neighbourhood of the points where the perturbation has been done. To do this we use Mehler–Heine type asymptotic formulae. As a consequence we can establish limit relations between the zeros of these polynomials and the ones of the Bessel functions of the first kind (or linear combinations of them). We do some numerical experiments to illustrate the results.