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  • UPPER BOUNDS ON POLYNOMIALS WITH SMALL GALOIS GROUP
    Mathematika (IF 0.875) Pub Date : 2020-09-20
    Robert J. Lemke Oliver; Frank Thorne

    When monic integral polynomials of degree n ⩾ 2 are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group isomorphic to S n . In particular, among such polynomials whose coefficients are bounded by B in absolute value, asymptotically ( 1 + o ( 1 ) ) ( 2 B + 1 ) n are irreducible and

    更新日期:2020-09-21
  • EXPECTED f‐VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF‐SPHERE
    Mathematika (IF 0.875) Pub Date : 2020-08-27
    Zakhar Kabluchko

    We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in R d . The expected f‐vector is expressed through the coefficients of the polynomial ( 1 + ( d − 1 ) 2 x 2 ) ( 1 + ( d − 3 ) 2 x 2 ) ( 1 + ( d − 5 ) 2 x 2 ) … . Also, we compute explicitly the expected f‐vector and the expected volume of

    更新日期:2020-08-27
  • INEQUALITIES BETWEEN MIXED VOLUMES OF CONVEX BODIES: VOLUME BOUNDS FOR THE MINKOWSKI SUM
    Mathematika (IF 0.875) Pub Date : 2020-08-26
    Gennadiy Averkov, Christopher Borger, Ivan Soprunov

    In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum P 1 + ⋯ + P d of d‐dimensional lattice polytopes is bounded from above by a function of order O ( m 2 d ) , where m is the mixed volume of the tuple ( P 1 , ⋯ , P d ) . This is a consequence of the well‐known Aleksandrov–Fenchel inequality. Esterov

    更新日期:2020-08-26
  • MANIFOLD MATCHING COMPLEXES
    Mathematika (IF 0.875) Pub Date : 2020-08-19
    Margaret Bayer, Bennet Goeckner, Marija Jelić Milutinović

    The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper, we completely characterize the pairs (graph, matching complex) for which the matching complex is a homology manifold, with or without boundary. Except in dimension two, all of these manifolds are spheres or balls.

    更新日期:2020-08-19
  • A HALES–JEWETT TYPE PROPERTY OF FINITE SOLVABLE GROUPS
    Mathematika (IF 0.875) Pub Date : 2020-08-17
    Vassilis Kanellopoulos, Miltiadis Karamanlis

    A conjecture of Leader, Russell, and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown, the “if” direction of their conjecture follows if all finite groups have a Hales–Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable

    更新日期:2020-08-17
  • WEISS'S QUESTION
    Mathematika (IF 0.875) Pub Date : 2020-08-10
    Péter Komjáth

    There is a function f : R + → { 0 , 1 } such that if F ( x , y ) = f ( d ( x , y ) ) ( d ( x , y ) is the distance between x and y), then there is no uncountable homogeneous set for F. If CH holds, we show that there is a similar coloring of R n with ℵ1 colors so that uncountable sets contain all colors.

    更新日期:2020-08-10
  • COMBINATORIAL INSCRIBABILITY OBSTRUCTIONS FOR HIGHER DIMENSIONAL POLYTOPES
    Mathematika (IF 0.875) Pub Date : 2020-08-07
    Joseph Doolittle, Jean‐Philippe Labbé, Carsten E. M. C. Lange, Rainer Sinn, Jonathan Spreer, Günter M. Ziegler

    For 3‐dimensional convex polytopes, inscribability is a classical property that is relatively well‐understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every f‐vector of 3‐polytopes, there exists an inscribable polytope with that f‐vector. For higher dimensional polytopes, much less is

    更新日期:2020-08-07
  • CONCRETE POLYTOPES MAY NOT TILE THE SPACE
    Mathematika (IF 0.875) Pub Date : 2020-07-16
    Alexey Garber, Igor Pak

    Brandolini et al. conjectured in (Preprint, 2019) that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in R 3 .

    更新日期:2020-07-16
  • SPECTRA OF DELAY EQUATIONS
    Mathematika (IF 0.875) Pub Date : 2020-07-08
    Luis Barreira, Claudia Valls

    We describe all possible forms of the Sacker–Sell spectrum for a nonautonomous linear delay equation. In particular, the spectrum may have infinitely many connected components either accumulating at minus infinity or at a finite value. We also show that to each connected component of the spectrum one can associate a subspace of initial conditions whose lower and upper Lyapunov exponents are contained

    更新日期:2020-07-08
  • AUTOMORPHIC SL2‐PERIODS AND THE SUBCONVEXITY PROBLEM FOR GL2 × GL3
    Mathematika (IF 0.875) Pub Date : 2020-07-03
    Aprameyo Pal, Carlos de Vera‐Piquero

    We prove a new (conditional) result toward the subconvexity problem for certain automorphic L‐functions for GL2 × GL3. This follows from the computation of new SL2‐period integrals associated with newforms f and g of even weight and odd squarefree level. The same computations also lead to a central value formula for degree six complex L‐series of the form L ( f ⊗ Ad ( g ) , s ) , extending previous

    更新日期:2020-07-03
  • AN OPTIMAL BOUND FOR THE RATIO BETWEEN ORDINARY AND UNIFORM EXPONENTS OF DIOPHANTINE APPROXIMATION
    Mathematika (IF 0.875) Pub Date : 2020-06-28
    Antoine Marnat, Nikolay G. Moshchevitin

    We provide a lower bound for the ratio between the ordinary and uniform exponents of both simultaneous Diophantine approximation to n real numbers and Diophantine approximation for one linear form in n variables. This question was first considered in the 1950s by Jarník who solved the problem for two real numbers and established certain bounds in higher dimension. Recently different authors reconsidered

    更新日期:2020-06-28
  • SATURATION FOR THE BUTTERFLY POSET
    Mathematika (IF 0.875) Pub Date : 2020-05-20
    Maria‐Romina Ivan

    Given a finite poset P , we call a family F of subsets of [n] P ‐saturated if F does not contain an induced copy of P , but adding any other set to F creates an induced copy of P . The induced saturated number of P , denoted by sat ∗ ( n , P ) , is the size of the smallest P ‐saturated family with ground set [n]. In this paper, we are mainly interested in the four‐point poset called the butterfly.

    更新日期:2020-05-20
  • LOW‐LYING ZEROS OF L‐FUNCTIONS FOR MAASS FORMS OVER IMAGINARY QUADRATIC FIELDS
    Mathematika (IF 0.875) Pub Date : 2020-05-13
    Sheng‐Chi Liu, Zhi Qi

    We study the 1‐ or 2‐level density of families of L‐functions for Hecke–Maass forms over an imaginary quadratic field F. For test functions whose Fourier transform is supported in ( − 3 2 , 3 2 ) , we prove that the 1‐level density for Hecke–Maass forms over F of square‐free level q , as N ( q ) tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke–Maass forms over

    更新日期:2020-05-13
  • ITERATIONS OF CURVATURE IMAGES
    Mathematika (IF 0.875) Pub Date : 2020-05-10
    Mohammad N. Ivaki

    We study the iterations of a class of curvature image operators Λ p φ introduced by Ivaki in [J. Funct. Anal. 271 (2016) 2133–2165]. The fixed points of these operators are the solutions of the L p Minkowski problems with the positive continuous prescribed data φ. One of our results states that if p ∈ ( − n , 1 ) and φ is even, or if p ∈ ( − n , − n + 1 ] , then the iterations of these operators applied

    更新日期:2020-05-10
  • EIGENVALUES OF THE FINSLER p‐LAPLACIAN ON VARYING DOMAINS
    Mathematika (IF 0.875) Pub Date : 2020-05-07
    Giuseppina di Blasio, Pier Domenico Lamberti

    We study the dependence of the first eigenvalue of the Finsler p‐Laplacian and the corresponding eigenfunctions on perturbation of the domain and we generalize a few results known for the standard p‐Laplacian. In particular, we prove a Frechét differentiability result for the eigenvalues, compute the corresponding Hadamard formulas and prove a continuity result for the eigenfunctions. Finally, we briefly

    更新日期:2020-05-07
  • NORMAL TILINGS OF A BANACH SPACE AND ITS BALL
    Mathematika (IF 0.875) Pub Date : 2020-05-06
    Robert Deville, Miguel García‐Bravo

    We show some new results about tilings in Banach spaces. A tiling of a Banach space X is a covering by closed sets with non‐empty interior, so that they have pairwise disjoint interiors. If, moreover, the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010, Preiss constructed a convex normal tiling of the separable

    更新日期:2020-05-06
  • HOMOGENEOUS COMPLETELY SIMPLE SEMIGROUPS
    Mathematika (IF 0.875) Pub Date : 2020-05-05
    Thomas Quinn‐Gregson

    A semigroup is completely simple if it has no proper ideals and contains a primitive idempotent. We say that a completely simple semigroup S is a homogeneous completely simple semigroup if any isomorphism between finitely generated completely simple sub‐semigroups of S extends to an automorphism of S. Motivated by the study of homogeneous completely regular semigroups, we obtain a complete classification

    更新日期:2020-05-05
  • BEYOND ARTIN'S CONJECTURE FOR CUBIC FORMS
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Miriam Sophie Kaesberg

    It is established that every cubic form in at least eight variables which can be realised as a diagonal form on a hyperplane has a non‐trivial p‐adic solution for all primes p.

    更新日期:2020-05-04
  • EXPONENTIAL MOMENTS OF THE ARGUMENT OF THE RIEMANN ZETA FUNCTION ON THE CRITICAL LINE
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Joseph Najnudel

    In this article, we give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta function on the critical line. Our result, which gives information on the fluctuations of the distribution of the zeros of ζ, has the same accuracy as the result obtained by Soundararajan (Ann. of Math. (2) 170(2) (2009), 981–993) for the moments

    更新日期:2020-05-04
  • VARIATIONAL INEQUALITIES FOR BILINEAR AVERAGES
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Honghai Liu, Hanbin Wang

    We establish variational inequalities for a class of bilinear averages of one variable, generalizing the variational inequalities for linear averages of Jones et al. As an application we obtain almost everywhere convergence for ergodic averages along cubes on a dynamical system.

    更新日期:2020-05-04
  • NOWHERE‐ANALYTIC SMOOTH CURVES WITH NON‐TRIVIAL ANALYTIC ISOTROPY
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Giuseppe Della Sala

    We study the smoothness properties of planar curves γ ⊂ R 2 , 0 ∈ γ , which are invariant under a local real‐analytic diffeomorphism ψ fixing the origin. Under certain conditions, depending on the first‐order jet (if the eigenvalues of d ψ ( 0 ) are not both of modulus one) or on a higher order jet (if ψ is tangent to the identity) of ψ and γ, we show that γ must be real analytic as soon as it is smooth

    更新日期:2020-05-04
  • EFFECTIVE l2 DECOUPLING FOR THE PARABOLA
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Zane Kun Li

    We make effective l 2 L p decoupling for the parabola in the range 4 < p < 6 . In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth‐order correlation of the integer solutions of the equation x 2 + y 2 = m in an extremal case. This proves unconditionally a result that was proven in Bombieri and Bourgain [Int. Math. Res. Not. IMRN 2015(11), 3343–3407]

    更新日期:2020-05-04
  • POISSON HYPERPLANE PROCESSES AND APPROXIMATION OF CONVEX BODIES
    Mathematika (IF 0.875) Pub Date : 2020-05-04
    Daniel Hug, Rolf Schneider

    A natural model for the approximation of a convex body K in R d by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in R d , and consider the random polytope Z K defined as the intersection of all closed halfspaces containing K that are bounded by hyperplanes of the process not intersecting K. If f is a functional on convex bodies, then for increasing intensities

    更新日期:2020-05-04
  • AN UPPER BOUND FOR DISCRETE MOMENTS OF THE DERIVATIVE OF THE RIEMANN ZETA‐FUNCTION
    Mathematika (IF 0.875) Pub Date : 2020-04-09
    Scott Kirila

    Assuming the Riemann hypothesis, we establish an upper bound for the 2kth discrete moment of the derivative of the Riemann zeta‐function at nontrivial zeros, where k is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments

    更新日期:2020-04-09
  • ON RANDOM APPROXIMATIONS BY GENERALIZED DISC‐POLYGONS
    Mathematika (IF 0.875) Pub Date : 2020-04-09
    Ferenc Fodor, Dániel I. Papvári, Viktor Vígh

    For two convex discs K and L, we say that K is L‐convex (Lángi et al., Aequationes Math. 85(1–2) (2013), 41–67) if it is equal to the intersection of all translates of L that contain K. In L‐convexity, the set L plays a similar role as closed half‐spaces do in the classical notion of convexity. We study the following probability model: Let K and L be C + 2 smooth convex discs such that K is L‐convex

    更新日期:2020-04-09
  • A NOTE ON ANTICHAINS IN THE CONTINUOUS CUBE
    Mathematika (IF 0.875) Pub Date : 2020-04-09
    Barnabás Janzer

    It is well known that an antichain in the continuous cube [0, 1]n must have measure 0. Engel, Mitsis, Pelekis and Reiher showed that in fact it must have ( n − 1 ) ‐dimensional Hausdorff measure at most n, and they conjectured that this bound can be attained. In this note we show that, for every n, such an antichain does indeed exist.

    更新日期:2020-04-09
  • SHARP BOUNDS FOR MULTILINEAR CURVED KAKEYA, RESTRICTION AND OSCILLATORY INTEGRAL ESTIMATES AWAY FROM THE ENDPOINT
    Mathematika (IF 0.875) Pub Date : 2020-04-09
    Terence Tao

    We revisit the multilinear Kakeya, curved Kakeya, restriction and oscillatory integral estimates that were obtained in a paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form R ε (or log O ( 1 ) R ) for some scale factor R. By further

    更新日期:2020-04-09
  • EMBEDDING SPANNING BOUNDED DEGREE GRAPHS IN RANDOMLY PERTURBED GRAPHS
    Mathematika (IF 0.875) Pub Date : 2020-04-02
    Julia Böttcher, Richard Montgomery, Olaf Parczyk, Yury Person

    We study the model G α ∪ G ( n , p ) of randomly perturbed dense graphs, where G α is any n‐vertex graph with minimum degree at least α n and G ( n , p ) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two

    更新日期:2020-04-02
  • ON UNIFORM CONTRACTIONS OF BALLS IN MINKOWSKI SPACES
    Mathematika (IF 0.875) Pub Date : 2020-04-02
    Károly Bezdek

    Let N balls of the same radius be given in a d‐dimensional real normed vector space, i.e., in a Minkowski d‐space. Then apply a uniform contraction to the centers of the N balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main

    更新日期:2020-04-02
  • DISCREPANCY OF A CONVEX SET WITH ZERO CURVATURE AT ONE POINT
    Mathematika (IF 0.875) Pub Date : 2020-04-02
    Bianca Gariboldi

    Let Ω be a convex body in R d and assume that ∂ Ω is smooth with everywhere positive Gaussian curvature except for one flat point of order γ ⩾ 2 . We consider the L p norm of the discrepancy with respect to translations and rotations of a dilated copy of the set Ω.

    更新日期:2020-04-02
  • REMARQUES SUR UNE SOMME LIÉE À LA FONCTION DE MÖBIUS
    Mathematika (IF 0.875) Pub Date : 2020-04-01
    Régis de la Bretèche, François Dress, Gérald Tenenbaum

    For integer n ⩾ 1 and real z ⩾ 1 , define M ( n , z ) : = ∑ d | n , d ⩽ z μ ( d ) , where μ denotes the Möbius function. Put L ( y ) : = exp { ( log y ) 3 / 5 / ( log 2 y ) 1 / 5 } ( y ⩾ 3 ) . We show that, for a suitable, explicit constant L > 0 and some constant c > 0 , we have S ( x , z ) = L x + O ( x / L ( 3 ξ ) c ) uniformly for x ⩾ 1 , ξ ⩽ z ⩽ x / ξ .

    更新日期:2020-04-01
  • ON THE MAGNITUDE AND INTRINSIC VOLUMES OF A CONVEX BODY IN EUCLIDEAN SPACE
    Mathematika (IF 0.875) Pub Date : 2020-03-30
    Mark W. Meckes

    Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is closely related to intrinsic volumes. Here we prove an upper bound for the magnitude of a convex body in Euclidean space in terms of its intrinsic volumes. The result is deduced from an analogous known

    更新日期:2020-03-30
  • ON POINTWISE CONVERGENCE OF SCHRÖDINGER MEANS
    Mathematika (IF 0.875) Pub Date : 2020-03-30
    Evangelos Dimou, Andreas Seeger

    For functions in the Sobolev space H s and decreasing sequences t n → 0 we examine convergence almost everywhere of the generalized Schrödinger means on the real line, given by S a f ( x , t n ) = exp ( i t n ( − ∂ x x ) a / 2 ) f ( x ) ; here a > 0 , a ≠ 1 . For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in H s when 0 < s < min { a / 4 , 1

    更新日期:2020-03-30
  • MEAN VALUES OF ARITHMETIC FUNCTIONS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS IN THE LARGE‐DEGREE LIMIT
    Mathematika (IF 0.875) Pub Date : 2020-03-30
    Ofir Gorodetsky

    A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as possible. We study this problem in the function field setting, and prove for a wide class of arithmetic functions (namely factorization functions), that

    更新日期:2020-03-30
  • THE DISTRIBUTION OF DIVISORS OF POLYNOMIALS
    Mathematika (IF 0.875) Pub Date : 2020-03-30
    Kevin Ford, Guoyou Qian

    Let F ( x ) be an irreducible polynomial with integer coefficients and degree at least 2. For x ⩾ z ⩾ y ⩾ 2 , denote by H F ( x , y , z ) the number of integers n ⩽ x such that F ( n ) has at least one divisor d with y < d ⩽ z . We determine the order of magnitude of H F ( x , y , z ) uniformly for y + y / log C y < z ⩽ y 2 and y ⩽ x 1 − δ , showing that the order is the same as the order of H ( x

    更新日期:2020-03-30
  • ROOTS OF TRIGONOMETRIC POLYNOMIALS AND THE ERDŐS–TURÁN THEOREM
    Mathematika (IF 0.875) Pub Date : 2020-03-27
    Stefan Steinerberger

    We prove, informally put, that it is not a coincidence that cos ( n θ ) + 1 ≥ 0 and the roots of z n + 1 = 0 are uniformly distributed in angle—a version of the statement holds for all trigonometric polynomials with “few” real roots. The Erdős–Turán theorem states that if p ( z ) = ∑ k = 0 n a k z k is suitably normalized and not too large for | z | = 1 , then its roots are clustered around | z | =

    更新日期:2020-03-27
  • ON THE WARING–GOLDBACH PROBLEM WITH ALMOST EQUAL SUMMANDS
    Mathematika (IF 0.875) Pub Date : 2020-03-27
    Juho Salmensuu

    We use transference principle to show that whenever s is suitably large depending on k ⩾ 2 , every sufficiently large natural number n satisfying some congruence conditions can be written in the form n = p 1 k + ⋯ + p s k , where p 1 , … , p s ∈ [ x − x θ , x + x θ ] are primes, x = ( n / s ) 1 / k and θ = 0.525 + ε . We also improve known results for θ when k ⩾ 2 and s ⩾ k 2 + k + 1 . For example

    更新日期:2020-03-27
  • SMALLNESS OF THE SET OF CRITICAL VALUES OF DISTANCE FUNCTIONS IN TWO‐DIMENSIONAL EUCLIDEAN AND RIEMANNIAN SPACES
    Mathematika (IF 0.875) Pub Date : 2020-03-27
    Jan Rataj, Luděk Zajíček

    We study how small is the set of critical values of the distance function from a compact (respectively, closed) set in the plane or in a connected complete two‐dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has “locally” (away from 0) bounded sum of square roots of lengths of gaps (components

    更新日期:2020-03-27
  • TRANSFERENCE THEOREMS FOR DIOPHANTINE APPROXIMATION WITH WEIGHTS
    Mathematika (IF 0.875) Pub Date : 2020-03-27
    Oleg N. German

    In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalise the corresponding inequalities that exist in the “nonweighted” case.

    更新日期:2020-03-27
  • AN APPLICATION OF THE MODULAR METHOD AND THE SYMPLECTIC ARGUMENT TO A LEBESGUE–NAGELL EQUATION
    Mathematika (IF 0.875) Pub Date : 2019-12-31
    Angelos Koutsianas

    We study the generalized Lebesgue–Nagell equation x 2 + 7 2 k + 1 = y n in integers x , y , k , n with k ⩾ 0 and n ⩾ 3 . We use the modular method, the multi–Frey technique and a symplectic argument. We also use a recent implementation of algorithms for solving Thue–Mahler equations by Adela Gherga for the cases n = 3 and 5. The combination of the previous methods allows us to get results for the case

    更新日期:2019-12-31
  • KHINTCHINE'S THEOREM WITH RANDOM FRACTIONS
    Mathematika (IF 0.875) Pub Date : 2019-12-05
    Felipe A. Ramírez

    We prove versions of Khintchine's theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions

    更新日期:2019-12-05
  • BOMBIERI–VINOGRADOV THEOREMS FOR MODULAR FORMS AND APPLICATIONS
    Mathematika (IF 0.875) Pub Date : 2019-12-05
    Peng‐Jie Wong

    In this article, we consider a prime number theorem for arithmetic progressions “weighted” by Fourier coefficients of modular forms, and we develop Siegel‐Walfisz type and Bombieri–Vinogradov type estimates for such a modular analogue. As an application, we have a Turán type estimate for modular forms asserting that for any δ > 0 and non‐CM normalised Hecke eigenform f, P f ( a , q ) ≤ q 2 + δ , with

    更新日期:2019-12-05
  • DIOPHANTINE APPROXIMATION WITH HARDY–LITTLEWOOD NUMBERS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Glyn Harman

    In this paper, we consider Diophantine approximation using numbers which are a sum of a prime and a square (Hardy–Littlewood Numbers). We use exponential sums and sieve methods to establish, for any irrational α and arbitrary real β, that there are infinitely many solutions to ∥ α n + β ∥ < n ε − 5 / 8 , n = p + m 2 , p a prime , where ∥ · ∥ denotes, as usual, distance to a nearest integer.

    更新日期:2019-11-26
  • NONUNIFORM EXPONENTIAL BEHAVIOR VIA EVOLUTION SEMIGROUPS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Luis Barreira, Liviu Horia Popescu, Claudia Valls

    We provide a unified framework to study the nonuniform exponential behavior of a dynamics using the theory of evolution semigroups. We emphasize that nonuniform hyperbolicity is ubiquitous in the context of ergodic theory: for an autonomous differential equation whose flow preserves a finite measure, the variational equations of almost all trajectories with nonzero Lyapunov exponents have a nonuniform

    更新日期:2019-11-26
  • THE FOURIER COEFFICIENTS OF Θ‐SERIES IN ARITHMETIC PROGRESSIONS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Guangwei Hu, Yujiao Jiang, Guangshi Lü

    In this paper, we first study the Fourier coefficients of Θ‐series in arithmetic progressions and its applications. Secondly, we introduce a rather simple argument to improve some results on the shifted convolution sums of the Fourier coefficients of a cusp form for S L ( 2 , Z ) or S L ( 3 , Z ) and a Θ‐series.

    更新日期:2019-11-26
  • SMOOTH SQUAREFREE AND SQUAREFULL INTEGERS IN ARITHMETIC PROGRESSIONS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Marc Munsch, Igor E. Shparlinski, Kam Hung Yau

    We obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve that of Balog and Pomerance. We also obtain an estimate on the smallest squarefull number in almost all residue classes modulo a prime p.

    更新日期:2019-11-26
  • EXPANDING POLYNOMIALS ON SETS WITH FEW PRODUCTS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Cosmin Pohoata

    In this note, we prove that if A is a finite set of real numbers such that | A A | = K | A | , then for every polynomial f ∈ R [ x , y ] , we have that | f ( A , A ) | = Ω K , deg f ( | A | 2 ) , unless f is of the form f ( x , y ) = g ( M ( x , y ) ) for some monomial M and some univariate polynomial g. This is sharp up to the dependence on K and the degree of f.

    更新日期:2019-11-26
  • METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Henry Adams, Johnathan Bush, Florian Frick

    Thickenings of a metric space capture local geometric properties of the space. Here, we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk–Ulam type. We use the combinatorial and geometric

    更新日期:2019-11-26
  • ON A QUESTION OF ALAN BAKER OVER ARBITRARY NUMBER FIELDS
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Abhishek T. Bharadwaj, Purusottam Rath

    For any positive integer q, it is a question of Baker whether the numbers L ( 1 , χ ) , where χ runs over the non‐trivial characters mod q, are linearly independent over Q . In this work, we consider the linear independence of the L ( 1 , χ ) values over number fields where χ runs over the non‐trivial characters modulo a prime p. The dimension over Q and over some specific number fields is known, thanks

    更新日期:2019-11-26
  • HARMONIC MAPS AND TWISTORIAL STRUCTURES
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    G. Deschamps, E. Loubeau, R. Pantilie

    We introduce the notion of Riemannian twistorial structure and show that it provides new natural constructions of harmonic maps.

    更新日期:2019-11-26
  • TENSOR AND CONVOLUTION DIRECT IMAGE OF ℓ‐ADIC SHEAVES
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Antonio Rojas‐León

    Given a Galois étale map of varieties π : Y → X and an ℓ‐adic sheaf or complex on Y, we study two cohomological operations: the tensor direct image and the convolution direct image (in the case of perverse sheaves), which give objects on X, and can be used to improve the estimates on some partial exponential sums and the number of rational points on certain varieties.

    更新日期:2019-11-26
  • ON THE OSCILLATION OF THE MODULUS OF THE RUDIN–SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Tamás Erdélyi

    In signal processing the Rudin–Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar and communication systems. In this paper we study the oscillation of the modulus of the Rudin–Shapiro polynomials on the unit circle. We also show that the Rudin–Shapiro polynomials

    更新日期:2019-11-26
  • EXTREMAL BANACH–MAZUR DISTANCE BETWEEN A SYMMETRIC CONVEX BODY AND AN ARBITRARY CONVEX BODY ON THE PLANE
    Mathematika (IF 0.875) Pub Date : 2019-11-26
    Tomasz Kobos

    We prove that if K , L ⊂ R 2 are convex bodies such that L is symmetric and the Banach–Mazur distance between K and L is equal to 2, then K is a triangle.

    更新日期:2019-11-26
  • THE MAXIMAL ORDER OF ITERATED MULTIPLICATIVE FUNCTIONS
    Mathematika (IF 0.875) Pub Date : 2019-07-30
    Christian Elsholtz, Marc Technau, Niclas Technau

    Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering

    更新日期:2019-07-30
  • AN EXPLICIT TWO‐SOURCE EXTRACTOR WITH MIN‐ENTROPY RATE NEAR
    Mathematika (IF 0.875) Pub Date : 2019-07-24
    Mark Lewko

    In 2005 Bourgain gave the first explicit construction of a two‐source extractor family with min‐entropy rate less than 1/2. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min‐entropy rate which was greater than 0.499. This remained essentially the state of the art until a 2015 breakthrough by Chattopadhyay and Zuckerman

    更新日期:2019-07-24
  • THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS
    Mathematika (IF 0.875) Pub Date : 2019-07-23
    Christoph Aistleitner, Thomas Lachmann, Niclas Technau

    We consider sequences of the form mod 1, where and where is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all in the sense of Lebesgue measure, we say that has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations

    更新日期:2019-07-23
  • FLAG AREA MEASURES
    Mathematika (IF 0.875) Pub Date : 2019-07-23
    Judit Abardia-Evéquoz, Andreas Bernig, Susanna Dann

    A flag area measure on an ‐dimensional euclidean vector space is a continuous translation‐invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector and a ‐dimensional linear subspace containing with . Using local parallel sets, Hinderer constructed examples of ‐covariant flag area measures. There is an explicit formula for his flag area measures

    更新日期:2019-07-23
  • A CERTAIN DIRICHLET SERIES OF RANKIN–SELBERG TYPE ASSOCIATED WITH THE IKEDA LIFT OF HALF‐INTEGRAL WEIGHT
    Mathematika (IF 0.875) Pub Date : 2019-06-17
    Shuichi Hayashida

    In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half‐integral weight. Here these Siegel cusp forms of half‐integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura

    更新日期:2019-06-17
  • THE POWER‐SAVING MANIN–PEYRE CONJECTURE FOR A SENARY CUBIC
    Mathematika (IF 0.875) Pub Date : 2019-05-22
    Sandro Bettin, Kevin Destagnol

    Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory 47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power‐saving error term for the two varieties respectively in with bihomogeneous coordinates and in with multihomogeneous coordinates defined by the same equation . We thus improve on recent

    更新日期:2019-05-22
  • ON SIMPLE ZEROS OF THE DEDEKIND ZETA‐FUNCTION OF A QUADRATIC NUMBER FIELD
    Mathematika (IF 0.875) Pub Date : 2019-05-22
    Xiaosheng Wu, Lilu Zhao

    We study the number of non‐trivial simple zeros of the Dedekind zeta‐function of a quadratic number field in the rectangle . We prove that such a number exceeds if is sufficiently large. This improves upon the classical lower bound established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math. 86 (1986), 563–576].

    更新日期:2019-05-22
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