Throughout this paper we assume that k is a given positive integer. As usual, B(x, r) denotes the closed ball with centre at x∈ℝk and radius r > 0. Let μ be a Radon measure on ℝk, that is, μ is locally finite and Borel regular. For s ≥ 0, the lower and upper s–dimensional densities of μ at x are denned respectively by

and

The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

The prototype of isoperimetric problems is to minimize the surface area of a convex body with given volume. The minimal body is naturally the suitable ball. The solution to this problem in the planar case was already known to the ancient Greeks. In the higher dimensional cases, the first proofs were provided with the help of Steiner's symmetrization method towards the end of the last century. Important later contributors are, among others, Minkowski, Blaschke, Hadwiger. By their work, the optimality of the ball has been also verified for a much wider class of sets (see [14]).

C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.

We determine all continuous translation invariant simple valuations on the space of convex bodies in d-dimensional Euclidean space.

Let K be a number field of degree nK = r1 + 2r2, a fixed integral ideal and the group of fractional ideals of K whose prime decomposition contains no prime factors of . Let

and be an arbitrary Groessencharaktere mod f as defined in [15]. Then and, for

where {λi} forms a basis for the torsion–free characters on whose value on any depends only on the that exists such that a = (α). Note that because of the choice in such an α we have that 1 for all units ε in K satisfying (mod ), ε>0. Also, x is a narrow ideal class character mod , that is, a character on

The sum or Minkowski addition of two subsets, A, B ⊂ En is defined by:

We recall that if S is a d - simplex then each facet and each vertex figure of S is a (d − 1)-simplex and S is a self-dual. We introduce a d-polytope P, called a d-multiplex, with the property that each facet and each vertex figure of P is a (d − 1)-multiplex and P is self-dual.

Recently Bombieri and Sperber have jointly created a new construction for estimating exponential sums on quasiprojective varieties over finite fields. In this paper we apply their construction to estimate hybrid exponential sums on quasiprojective varieties over finite fields. In doing this we utilize a result of Aldolphson and Sperber concerning the degree of the L-function associated with a certain exponential sum.

The distribution of squarefree binomial coefficients. For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

In this paper we prove that a maximal common summand of two compact convex sets in R2 is unique up to translation.

In [3] the authors introduced the notion of a completely 0-simple semigroup of quotients. This definition has since been extended to the class of all semigroups giving a definition of semigroups of quotients which may be regarded as an analogue of the classical ring of quotients. When Q is a semigroup of quotients of a semigroup S, we also say that S is an order in Q.

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.

In 1984 Heath-Brown [5] proved the following conjecture of Erdős and Mirsky [2] (which seemed at one time as hard as the twin prime problem):

“There exist infinitely many integers n for which d(n) = d(n + 1).”

Let R be a real closed field and a variety of real dimension k′ which is the zero set of a polynomial Q∈R[X1,…, Xk] of degree at most d. Given a family of s polynomials = {P1,…, Ps}⊂R[X1,…,Xk] where each polynomial in has degree at most d, we prove that the number of cells defined by over is (O(d))k Note that the combinatorial part of the bound depends on the dimension of the variety rather than on the dimension of the ambient space.

This paper studies higher dimensional analogues of the Tamari lattice on triangulations of a convex n-gon, by placing a partial order on the triangulations of a cyclic d-polytope. Our principal results are that in dimension d≤3, these posets are lattices whose intervals have the homotopy type of a sphere or ball, and in dimension d≤5, all triangulations of a cyclic d-polytope are connected by bistellar operations.

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

Let Δn = {(x, y): x, y are integers 1 ≤ x, y ≤ n} be the n x n square array of integer lattice points in the plane.

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.