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

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.)

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 .

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.

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.

Let

and let p denote any prime. The p-content vp(f) of f is denned by

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.

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 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

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 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}.

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

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).

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.

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.

The Brianchon-Gram and Sommerville theorems on angle-sums for convex polytopes and polyhedral cones are here shown to be particular cases of an angle-sum relation for general polyhedral sets. The new relation is proved on the level of an equidissectability theorem, and this approach yields yet other angle-sum relations, including a different generalization of the Brianchon-Gram theorem. Further results extend, again to equidissections, earlier angle-sum relations of the author and others.

The purpose of this paper is to define and study a natural rank function which associates to each differentiable function (say on the interval [0,1]) a countable ordinal number, which measures the complexity of its derivative. Functions with continuous derivatives have the smallest possible rank 1, a function like x2 sin (x−1) has rank 2, etc., and we show that functions of any given countable ordinal rank exist. This exhibits an underlying hierarchical structure of the class of differentiable functions, consisting of ω1, distinct levels. The definition of rank is invariant under addition of constants, and so it naturally assigns also to every derivative a unique rank, and an associated hierarchy for the class of all derivatives.