Ramsey's theorem implies that every function f:0, 1 ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r ). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

Let Ω be a compact metric space of non-σ-finite Λh-measure, where Λh denotes the Hausdorff measure corresponding to some continuous increasing real function h(t), denned for t ≥ 0 and with h(t) > 0 and h(t) > 0 for t > 0. C. A. Rogers has drawn attention to two problems that have remained unsolved for some years, and which seem to present considerable difficulty:

I. Does Ω necessarily have subsets of finite positive Λh-measure?

II. Does Ω. necessarily contain a system ofdisjoint closed subsets each of non-σ-finite Λh-measure ?

The answers are affirmative when Ω is a subset of a Euclidean space (see [1], [2], [3], [4], [5]); recently D. G. Larman ([8], [9]) was able to extend each of these results to the case of the “finite-dimensional” compact metric spaces that he introduced in [7]. Both of his proofs depend on the detailed structure of such spaces; the aim of the present paper is to establish a general theorem, from which it follows in particular that Larman's second result is a consequence of his first, and that an affirmative answer to I would imply an affirmative answer to II. (Since it seems likely that I must be answered negatively in general, and II affirmatively, the present theorem may well be of only ephemeral significance; the nature of the proof reinforces this expectation.)

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k2r + 2. In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r2k2+(2/(c − 2)) if r ≠ 1 and p > k2+(2/(c − 1)) if r = 1.

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

The following theorem, due to Artin [1], is one of a number of statements often referred to as Chebotarev density theorem.

Let A⊆ℕ, let p be a prime and w a word over ℤ pℤ ending with a non-zero digit. The relationship is investigated between the density of A. the length of w and the density of the set of numbers n for which the base p expansion of ends with w0n for some a ∈ A. Also considered is the analogous problem on Pascal's triangle. This leads in particular to answering a question of Granville and Zhu [7] regarding the asymptotic frequency of sums of 3 squares in Pascal's triangle.

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.

In this note we generalize the following result of Sawyer [5]:

Theorem 1. There is a function ψ on ℝ such that, whenever g is a real-valued Borel measurable function on (a subset of) ℝ. × ℝn-1 with the property that y ↦ g(y, t) is C1 for a.e. t, the set

has measure zero.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.