It was remarked by Liouville in 1844 that there is an obvious limit to the accuracy with which algebraic numbers can be approximated by rational numbers; if α is an algebraic number of degree n (at least 2) then

for all rational numbers h/q, where A is a positive number depending only on α.

The method devised by Hardy and Littlewood for the solution of Waring's Problem, and further developed by Vinogradov, applies quite generally to Diophantine equations of an additive type. One particular result that can be proved by this method is that if a1, …, as are integers not all of the same sign, and if s is greater than a certain number depending only on k, the Diophantine equation

has infinitely many solutions in positive integers x1, …, xs, provided that the corresponding congruence has a non-zero solution to every prime power modulus. A result of a similar kind holds if on the right of (1) there stands an arbitrary integer in place of 0.

Two finite real functions ƒ(x) and g(x), defined for — ∞ < x < ∞, are said to be Riemann equivalent if |ƒ(x)—g(x)| has a zero Riemann integral over every finite interval; we then write ƒ~g or

N. G. de Bruijn conjectured that if ƒ(x+h)~ƒ(x) for every real number h, then ƒ~c where c is a constant; this was proved by P. Erdös [1]. In this note we associate with an arbitrary function ƒ the additive group (ƒ) of all numbers h which make ƒ(x)~ƒ(x+h), i.e. which make

In a recent paper [4], I introduced the notion of recursive formal Lie groups (of infinite dimension) over a field of characteristic p > 0, and studied a particular class of such groups, the groups of hyperexponential type; these can be characterized either as being (recursively) isomorphic to a special group of that class, the hyperexponential group, or by simple conditions on their Lie hyperalgebra. An interesting example of a group of that class is the additive Witt group W, whose “infinitesimal” structure can therefore be considered as known, at least “up to an isomorphism”. However, the intrinsic importance of the Witt group (which, as well known, is the “formalization”, so to speak, of the additive group of a p-adic field) leads one to think that it may be worth while to study in greater detail that group itself, instead of being content with the mere existence of an unspecified isomorphism with the hyperexponential group. This is what we intend to do here; it turns out that, although it seems hopeless to write down explicitly the group law of the Witt group, the multiplication table of its hyperalgebra is, on the contrary, as simple as one could hope, and is, in fact, identical to that of the hyperalgebra of the hyperexponential group (although the two groups are distinct). Moreover, this leads to a new and quite unexpected definition of the Witt group, which links it still closer to the hyperexponential group, and provides a well-determined isomorphism between the two groups.

The main object of this paper is to show that an indefinite quadratic form in four or more variables with integral coefficients represents all integers except those excluded by congruence considerations. The subject of the representation of integers by quadratic forms is one on which little is known, and I believe this result to be new, although I can deduce it in an elementary way from the classical theory.

The idea of a geometry in which the coordinates are elements of a linear algebra, instead of the conventional field, goes back to C. Segre. Most of the subsequent work seems to have been done by N. Spampinato who developed some general results and applied them particularly to the case of an algebra of dual numbers defined over the complex field; in general, his aim appears to have been the study of algebraic varieties in the new kind of space.

The main case of Siegel's theorem on algebraic curves may be stated as follows:

THEOREM 1. Let

be an irreducible algebraic curve of genus g≥1, ƒ(x, y) being a polynomial with algebraic coefficients. Let K be an algebraic field of finite degree over the rational field; let o be the ring of integers in K; and let j be a positive rational integer. Then there are at most finitely many points (x, y) on ℭ for which jxεο and yεK.

A well-known method of calculating an approximate value of an eigenvalue λ of a self-adjoint operator H employs an approximation, say w, to the corresponding eigenfunction ψ. Since

the exact value of λ is given by

In a recent book, L. H. Loomis has obtained the “Bohr compactification” of a topological group, in terms of almost periodic functions, by applying the representation theory of commutative B-algebras. It is simpler, and perhaps more natural, to consider this matter from the point of view of comparative topology; we can then obtain a more general result, in that the discussion is no longer restricted to the case of numerically valued (or even vector-valued) functions.

An analogy is drawn between the slow steady rotation of a solid of revolution about its axis in a viscous fluid and the motion, with constant velocity, of the solid along its axis in a perfect fluid. The rotation of a sphere in viscous fluid is then discussed using a method due in this case to Bickley [1] and here further developed.

In this note it is shown that

(i) there is a plane convex set of minimal width λ which does not contain any convex set whose width is constant and greater than λ/(3—31/2),

(ii) every plane convex set of minimal width λ contains a convex set whose width is constant and is equal to λ/(3—31/2).

In several of his papers, Mordell has developed a method which, in certain cases, leads to an inequality connecting the critical determinant of an n dimensional star body with that of a related n—1 dimensional star body. The purpose of this paper is to exhibit the underlying principle in a general form and to show that the same principle can sometimes be carried further.

Certain results ([7], [8], [10], [11]) suggest that there should be some principle of duality in homotopy theory. Among other things one is led to expect that cohomotopy groups will appear as dual to homotopy groups. But the fact that a cohomotopy group πn(X), unlike πn(X), is only defined if dim X ≤ 2n—2 is a serious obstacle to the formulation of such a principle. However, the set of S-maps (i.e.S-homotopy classes [11]) X → Y is a group for every pair of spaces X, Y. Therefore, this difficulty does not appear in S-theory [11].

Consider a set of n points lying in a square of side 1. Verblunsky has shown that, if n is sufficiently large, there is some path through all n points whose length does not exceed (2·8n)1/2+2. L. Fejes Tóth has drawn attention to the case when the n points consist of all points of a regular hexagonal lattice lying in the unit square, in which case the length of the shortest path is easily seen to be asymptotically equal to

Let →(X) be a function of the n variables (X) = (X1, …, Xn) defined for all real (X). A fundamental problem in the theory of Diophantine approximation is to prove the existence of real numbers (X) ≡ (x) (mod 1), where (x) = (x1, …, xn) is any given set of real numbers, for which

The analysis of the intervals which arise between events occurring randomly in time is a problem which is both interesting and important statistically. Two distinct types of data may arise: either the period of time during which the events are observed may be fixed or the number of intervals may be fixed. It may happen that the intervals between pairs of events, close in time, cannot be accurately measured. It is thus necessary to consider the lengths of intervals ordered according to their magnitudes. We derive here functions of these ordered interval lengths which may be used as a basis for tests for randomness of events occurring in a fixed period of time. The mathematical formulation of this problem is in terms of the classical problem of intervals between points on a line.

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequality

has only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

On page 2 of my paper “Rational approximations to algebraic numbers” [Mathematika 2 (1955), 1–20], I referred to possible generalizations of my theorem, and mentioned that I had proved the following:

Let α be any algebraic number, not rational. If the inequality

is satisfied by infinitely many algebraic numbers β of degree g, then κ ≤ 2g.