Roy O. Davies

Partitioning the plane into denumerably many sets without repeated distances

Ceder(1) proved (assuming the axiom of choice, as we do throughout this paper) that the Euclidean plane can be partitioned into ℵ 0 sets none of which contains an equilateral triangle; indeed he proved that given any denumerable set of triangles, the plane can be partitioned into ℵ 0 sets, none containing a triangle similar to one of the given triangles. Erdős and Kakutani(2) proved that the continuum hypothesis implies that the real line can be partitioned into ℵ 0 , rationally independent sete, and that the existence of such a partition implies the continuum hypothesis. Erdős asked (private communication) whether there is a partition of the plane into ℵ 0 sets not containing an isosceles triangle, or more generally in which any four points determine six different distances. Assuming the continuum hypothesis, it will be shown here (Theorem 1) that a, partition of the latter kind does exist. (I communicated this result to Erdős and others some years ago, but subsequently noticed that the argument was incomplete.) Conversely (Theorem 2) the existence of such a partitition, even for the line, implies the continuum hypothesis. This strengthening of the converse half of the Erdős-Kakutani theorem is proved by what is essentially their method (actually in a rather simplified form).

Separate approximate continuity implies measurability

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in ( x, y ), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Misiks question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

A direct method for completing eigenproblem solutions on a parallel computer

Abstract The computation of eigenvalues and eigenvectors of a real symmetric matrix A with distinct eigenvalues can be speeded up at the end of the Jacobi process when the off-diagonal elements have become sufficiently small for A to be regarded as a perturbation of a diagonal matrix. A leading-order approximation to the eigensolution is calculated by formulae particularly suitable for the distributed array processor (DAP). A single application of this direct method reduces A to diagonal form and is asymptotically equivalent to an entire sweep of the Jacobi method.

A property of Hausdorff measure

From the fact that Hausdorff s -dimensional measure is a regular Caratheodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result: Theorem A. If {E n } is any increasing sequence of sets, then ∧ s E n → ∧ s (Σ E n ) as n → ∞. Since ∧ s X is denned (for every set X ) as , the problem arises whether for every positive δ and every increasing sequence of sets one can prove Now there are four possible ways of defining ; it is the lower bound of taken over coverings Σ U r of X by convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be (i) open and of diameter less than δ, (ii) closed and of diameter less than δ, (iii) open and of diameter not exceeding δ, or (iv) closed and of diameter not exceeding δ.

Sets which are null or non-sigma-finite for every translation invariant measure

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure . This answers a question asked me by C. Dellacherie.

Lebesgue density influences Hausdorff measure; large sets surface-like from many directions

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets F n , whose intersection has some required property; it is desired that ∩ F n shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each F n+1 sufficiently “dense” in F n . Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝ k have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k ) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrands result [4] for the case k = 2.

A divided-difference characterization of polynomials over a general field

Summary. It is proved that, for an arbitrary field K not of characteristic 2 and arbitrary

Complete lattices and the generalized Cantor theorem

n \ge 2

On a selection problem for a sequence of finite sets

, if functions f : K→K and h : K→K satisfy f [x1, . . . , xn] = h (x1 + . . . + xn) whenever x1, . . . , xn are distinct elements of K, then f is equal to a polynomial of degree at most n over K. (Here f [x1, . . . , xn] denotes the divided difference of f at the distinct points x1, . . . , xn.) The case of a field of characteristic 2 is also considered.

A note on linear derivates of measurable functions

Cantors Theorem is generalized to a theorem on partially ordered sets. We shall show that every monotone mapping of a complete lattice into itself has a point of left continuity and a point of right continu- ity. From this result we derive an extension of a theorem of Gleason and Dilworth (2) which in turn can be regarded as a generalization of the classical theorem of Cantor stating that the cardinal of a set is less than the cardinal of its power-set. As a corollary it follows that if E and F are partially ordered sets then the cardinal power FE is not a homomorphic image of E unless \F\ =1. This result answers a question of F. W. Lawvere which provided the stimulus for our investigation. 1. A continuity theorem for complete lattices. If E and F are partially ordered sets then a mapping (a) = V (a) = A d>(x).