Bulletin of the London Mathematical Society - current issue

On the structure of steps of three-term arithmetic progressions in a dense set of integers

Candela, P. — Wed, 20 Jan 2010 08:40:25 PST

We use recent results in quadratic Fourier analysis to examine the additive structure of the set of steps (or common differences) of three-term arithmetic progressions in a general subset of [N]={1, 2, ..., N} of fixed positive density. In particular, combining the decomposition results of Gowers and Wolf with the recurrence results of Green and Tao, we show that if A [N] has density > 0, then, for some positive constant c = c(), the set of steps of three-term arithmetic progressions in A contains an arithmetic progression of length at least c(log log N)^c. This improves on the estimate of shape (log log log log log N) that one can obtain by a straightforward application of Gowers’ bounds for Szemerédi's theorem.

Relations between stable dimension and the preimage counting function on basic sets with overlaps

Mihailescu, E., Urbanski, M. — Wed, 20 Jan 2010 08:40:25 PST

In this paper we study non-invertible hyperbolic maps f and the relation between the stable dimension (that is, the Hausdorff dimension of the intersection between local stable manifolds of f and a given basic set ) and the preimage counting function of the map f restricted to the fractal set . The case of diffeomorphisms on surfaces was considered in [ A. Manning and H. McCluskey, ‘Hausdorff dimension for horseshoes’, Ergodic Theory Dynam. Systems 3 (1983) 251–260], where thermodynamic formalism was used to study the stable/unstable dimensions. In the case of endomorphisms, the non-invertibility generates new phenomena and new difficulties due to the overlappings coming from the different preimages of points, and also due to the variations of the number of preimages belonging to (when compared with [E. Mihailescu and M. Urbanski, ‘Estimates for the stable dimension for holomorphic maps’, Houston J. Math. 31 (2005) 367–389]). We show that, if the number of preimages belonging to of any point is less than or equal to a continuous function (·) on , then the stable dimension at every point is greater than or equal to the zero of the pressure function t -> P(t^s–log (·)). As a consequence we obtain that, if d is the maximum value of the preimage counting function on and if there exists x with the stable dimension at x equal to the zero t_d of the pressure function t -> P(t ^s – log d), then the number of preimages in of any point y is equal to d, and the stable dimension is t_d everywhere on . This has further consequences to estimating the stable dimension for non-invertible skew products with overlaps in fibers.

Upper bounds for moments of {zeta}'({rho})

Milinovich, M. B. — Wed, 20 Jan 2010 08:40:25 PST

Assuming the Riemann hypothesis, we obtain an upper bound for the 2kth moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of (s) for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments.

Defining the smooth points of a quotient in polynomially bounded o-minimal structures

Bew, D. — Wed, 20 Jan 2010 08:40:25 PST

Working in a polynomially bounded o-minimal structure over R, we prove a ‘Tamm property’ for quotients of smooth definable functions. For sufficiently large N, the smooth points and the C^N points of the quotient coincide. We use standard properties of these structures and a consequence of resolution of singularities.

Homology and cohomology operations in terms of differential operators

Brunetti, M., Ciampella, A., Lomonaco, L. A. — Wed, 20 Jan 2010 08:40:25 PST

We consider some actions of the universal Steenrod algebra Q on the graded algebra of finite Laurent series L(n) = F ₂[x₁^{± 1}, ... , x_n^{± 1}] compatible with the familiar action of the ordinary Steenrod algebra A on H* ((RP)ⁿ, F₂). The induced actions of the lambda algebra and the Dyer–Lashof algebra R on L(n)^– = F ₂[x₁^{– 1}, ... , x_n^{– 1}] are also studied. It turns out that the negative generators of Q do not act as differential operators on L(n), if the Cartan formula holds. We also prove that neither nor R are differential operator algebras when they act non-trivially on L(n)^–.

Embedded o-minimal structures

Hasson, A., Onshuus, A. — Wed, 20 Jan 2010 08:40:25 PST

We prove the following two theorems on embedded o-minimal structures:

Theorem 1. Let M N be o-minimal structures and let M* be the expansion of M by all traces in M of 1-variable formulas in N, that is all sets of the form (M, a) for a N and (x, y) L(N). Then, for any N-formula (x₁, ..., x_k), the set (M^k) is M*-definable.

Theorem 2. Let N be an ₁-saturated structure and let S be a sort in N^eq. Let S be the N-induced structure on S and assume that S is o-minimal. Then S is stably embedded.

The asymptotic formula for localized solutions in Waring's problem and approximations to Weyl sums

Daemen, D. — Wed, 20 Jan 2010 08:40:25 PST

An asymptotic formula is established for the number of representations of a natural number as the sum of s positive kth powers that are ‘almost equal’, in a quantitative sense. The results have unexpected applications to the distribution of values of classical Weyl sums.

Geometric remarks on the level curves of harmonic functions

De Carli, L., Hudson, S. M. — Wed, 20 Jan 2010 08:40:25 PST

Suppose that u is a nonconstant harmonic function on the plane. By the maximum principle, its zero set Z does not contain any simple closed curve. This paper provides bounds on the curvature of Z, and other conditions on Z, which imply that u is of some special type, such as polynomial, linear, or even constant. These theorems resemble unique continuation results, which often apply to various classes of nonharmonic functions. For such functions, Z may contain a closed curve, but a lower bound is established for the area of the enclosed region.

Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

Colbois, B., Dryden, E. B., El Soufi, A. — Wed, 20 Jan 2010 08:40:25 PST

We give upper bounds for the eigenvalues of the Laplace–Beltrami operator of a compact m-dimensional submanifold M of R^m+p. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of M with a p-plane in a generic position (transverse to M), or an invariant that measures the concentration of the volume of M in R^m+p. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand we show that, even for hypersurfaces (that is, when p = 1), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for m ≥ 3) the differential structure.

Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs

Chousionis, V., Mattila, P. — Wed, 20 Jan 2010 08:40:25 PST

We shall consider the truncated singular integral operators and related maximal operators . We shall prove for a large class of kernels K and measures µ and that if µ and are separated by a Lipschitz graph, then is bounded for 1 < p < . We shall also show that the truncated operators converge weakly in some dense subspaces of L²(µ) under mild assumptions for the measures and the kernels.

Universal series in {cap}p>1{ell}p

Koumandos, S., Nestoridis, V., Smyrlis, Y.-S., Stefanopoulos, V. — Wed, 20 Jan 2010 08:40:25 PST

In this paper an abstract condition is given yielding universal series defined by sequences a = {a_j}_j=1 in _p>1^p but not in ¹. We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann -function, or a fundamental solution of a given elliptic operator in R with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in R simultaneously with respect to all -finite Borel measures in R. Stronger results are obtained by using universal Dirichlet series.

Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups

Casals-Ruiz, M., Kazachkov, I., Remeslennikov, V. — Wed, 20 Jan 2010 08:40:25 PST

We show that graph products of finite abelian groups are elementarily equivalent if and only if they are -equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.

K3 spectra

Szymik, M. — Wed, 20 Jan 2010 08:40:25 PST

The notion of a K3 spectrum is introduced in analogy with that of an elliptic spectrum and it is shown that there are ‘enough’ K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the moduli stack in X with respect to the etale topology, and similarly for the ring of formal functions on the formal deformation space.

On simultaneously badly approximable numbers

Moshchevitin, N. G. — Wed, 20 Jan 2010 08:40:25 PST

Let , β, [0, 1], + β = 1 and It is proved that for different (₁, β₁), (₂, β₂), ₁ + β₁ = ₂ + β₂ = 1 and small enough This result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.

Commutators on {ell} ${infty}$

Dosev, D., Johnson, W. B. — Wed, 20 Jan 2010 08:40:25 PST

The operators on that are commutators are those not of the form I + S with != 0 and S strictly singular.

Raoul Harry Bott, FRS, 1923-2005

Atiyah, M. — Wed, 20 Jan 2010 08:40:25 PST

The Novikov conjecture, geometry and algebra * (OWS - Oberwolfach Seminars 33) * By Matthias Kreck and Wolfgang Luck: 266 pp., {euro}38.00 (CHF64.00), ISBN 3-7643-7141-2 * (Birkhauser, Basel, 2005)

Ranicki, A. — Wed, 20 Jan 2010 08:40:25 PST

Asymptotic analysis of random walks: * Heavy-tailed distributions * (Encyclopedia of Mathematics and Its Applications 118) * By A. A. Borovkov and K. A. Borovkov (Tr. O. B. Borovkova): 625 pp., {pound}95.00 * (US$190.00), ISBN 978-0-521-88117-3 * (Cambridge University Press, Cambridge, 2008)

Bingham, N. H. — Wed, 20 Jan 2010 08:40:25 PST

General theory of Lie groupoids and Lie algebroids * (London Mathematical Society Lecture Note Series 213) * By Kirill C. H. Mackenzie: xxxviii+501 pp., {pound}50.00 (US$90.00) (LMS members' price {pound}37.50 (US$67.50)), ISBN 0-521-49928-3 * (Cambridge University Press, Cambridge, 2005)

Voronov, T. — Wed, 20 Jan 2010 08:40:25 PST