Let

be a quadratic form with integral coefficients, and suppose the equation

has a solution in integers x1…, xn, not all 0. It was proved by Cassels [2] that there is such a solution, which satisfies the estimate

where F = max|fij|. It was later observed by Birch and Davenport [1] that the result can be stated in a slightly more general form. Let

be a quadratic form which assumes only integral values at the points (x1 …, x2) of an n-dimensional lattice Λ of determinant Δ. Suppose there is some point of Λ, other than the origin, at which ø = 0. Then there is such a point for which also

where Φ = max |øij|.

Let be a finite sequence of positive integers. If we want to show that contains primes, or at least almost-primes (i.e. numbers with few prime factors), sieve methods give better results if weights of a certain kind are attached to the elements of .

By a plane tree or rooted plane tree is meant a realization of a tree or rooted tree by points and arcs in the plane. By an isomorphism between two plane trees or rooted plane trees is meant an isomorphism in the usual sense for such trees which preserves the clockwise cyclic order of the edges about each node. In [2] Harary, Prins, and Tutte demonstrate how Polya's Theorem may be used to obtain formal expressions for the enumerating functions for unrooted, rooted, and other species of plane trees. In the process they obtain the explicit formula

for the number of nonisomorphic planted plane trees with n ≥ 1 edges. (A planted tree is a rooted tree with root at a node of degree one.)

It is the purpose of this note to draw attention to some connections between the topics mentioned in the title. These allow familiar results from one area of mathematics to be usefully exploited in another, to the mutual benefit of both. Moreover, the basic ideas suggest possible generalizations whose examination should prove worthwhile. In this introduction we give a brief account of the results to be derived and discussed in more detail in the later sections.

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).

Let p be an odd prime and denote by [p] the field of p elements. Let C = C(n, p) be the set of points x = (x1 …, xn) of Zn (n ≥ 1) satisfying

The Littlewood conjecture states that for all (α,β)∈ℝ2. We show that with the additional factor of log q⋅log log q, the statement is false. Indeed, our main result implies that the set of (α,β) for which is of full dimension.

A classical problem in the theory of convex polytopes is the enumeration of the distinct combinatorial types of d-polytopes with υ vertices (υ ≥ d + 1). Following Grünbaum [1] c(υ, d) will denote the number of such types. Apart from the general results c(d + l, d)= 1 and c(d + 2, d) = [¼d2], which may be established by elementary arguments, the only other known values of c(υ, d) are for small values of υ and d. These have been determined empirically; details of the most recent results are contained in [2].

In this paper the Hausdorff dimension of systems of real linear forms which are simultaneously small for infinitely many integer vectors is determined. A system of real linear forms,

where ai, xij∈ℝ, 1 ≤i≤m, 1≤j≤n will be denoted more concisely as

where a∈⇝m, X∈ℝmn and ℝmn is identified with Mm × n(ℝ), the set of real m × n matrices. The supremum norm of any vector in k dimensional Euclidean space, ℝk will be denoted by |v|. The distance of a point a from a set B, will be denoted by dist (a, B) = inf {|a − b|: b ∈ B}.

In recent papers on fractals attention has shifted from sets to measures [1, 5, 10]. Thus it seems interesting to know whether results for the dimension of sets remain valid for the dimension of measures. In the present paper we derive estimates for the dimension of product measures. Falconer [3] summarizes known results for sets and Tricot [8] gives a complete description in terms of Hausdorff and packing dimension. Let dim and Dim denote Hausdorff and packing dimension. If then

The object of this paper is to generalize to infinite CW complexes the known pull-back theorems or fracture lemmas concerning maps from finite CW complexes to localizations and completions.

If L is a set of disjoint closed line segments in En, let E(L) denote the end set of L, i.e. the set of end points of members of L.

In [1, 2] V. L. Klee and M. Martin proved the following lemma:

IfLis a disjoint set of closed line segments in E2 such that E(L) is compact, then E(L) has zero 2-measure.

Let ci, and di, (1≤i≤s) be rational integers, and k and n be natural numbers. We shall consider the solubility over the p-adic integers ℤp of the pair of additive equations

In the present paper we consider a one-dimensional local ring Q with maximal ideal tn and residue field K = Q/m. It will be assumed that not every element of mis a zero-divisor but no other restricting hypothesis will be made. In particular Q and K may have unequal characteristics and K may be finite.

The Hausdorff dimension has been used for many years for assessing the sizes of sets in Euclidean and other metric spaces, see, for example, [1,2,5,6,8,10]. However, different sets with the same Hausdorff dimension may have very different characteristics, for example, a straight line segment in ℝ2 and the Cartesian product in ℝ2 of two suitably chosen Cantor sets in ℝ will both have Hausdorff dimension 1. In this paper we develop a measure-theoretic method of distinguishing between the sets of such pairs.

There are at least two indices used to measure the size of bounded sets of ℝn of zero measure—Hausdorff dimension (see [4] for a definition), and the density index [7].

The paper deals with the problem of estimating the distance, in radial or Hausdorff metrics, between two centred star bodies of Rd, d≤3, in terms of the distance between the corresponding intersection bodies.

The paper shows that no origin-symmetric convex polyhedron in R3 is the intersection body of a star body. It is shown also that every originsymmetric convex body in Rd, for d = 3 and 4, can be seen as the intersection body of a star-shaped set whose radial function satisfies conditions related to suitable non-integer Sobolev classes.

§1. Introduction. In this note we shall discuss a certain dichotomy concerning the pointwise convergence of sequences of analytic sets in completely metrizable separable spaces (Proposition 2.1). The dichotomy is closely related to some reasoning due to W. Szlenk [Sz]; we comment on this in Section 5.1.

Let f(x) be a trigonometric polynomial with N (≥2) non-zero coefficients of absolute value not less than 1. In this paper it is proved that the L1 norm of f exceeds a fixed positive multiple of (logN/log log N)½. This result improves a previous one due to H. Davenport and P. J. Cohen (the same bound with exponent ¼).