In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let

be s non-negative functions of the positive integer n, and let

Let α, β, γ, δ be real numbers with Δ = |αδ −βγ| > 0, and let ξ, η denote the linear forms

The results of this paper have as their background two concepts fundamental in the theory of non-associative algebras (i.e. algebras in which the associative law of multiplication is not assumed). These concepts are:

(1) Isotopy,

(2)The three nuclei λ, μ, ρ of an algebra .

Let ν(M) denote the number of integers N in the range (0 ≤ N ≤ M) for which

where {y} denotes the fractional part of y. It is a convenient but trivial restriction to assume always that θ and φ lie between 0 and 1. There is a well-known result, which will be called ‘Weyl's Theorem’, that if θ is irrational,

1. The aim of this note is to prove the

Theorem. Let

where the λnare real and

and let

Then

A similar result holds for infinite seriesconverging uniformly in [−T, T].

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.

Let

be w sequences of differentiable functions defined in the ranges

where aj, bj may be − ∞, + ∞ respectively. For each j and each m ≠ n let

be monotonic and let there be a constant K independent of m, n, j, θj such that

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:

Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:

(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differences

with respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;

if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfying

is measurable for any realbi, tiand has measure F(t; b).

The counterfeit coin problems consist in detecting and locating the anomalous weights of counterfeit coins amongst a set of genuine coins. Suppose that altogether there are n coins, of which at least n − k are known to be genuine. We do not know which coins are genuine; but we do know that all genuine coins have the same (unknown) weight. The weights of the remaining k coins are unspecified and possibly all different.

It is well known that, if a trigonometric series

converges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

A. S. Besicovitch has defined the dimension of a point-set X in n-dimensional Euclidean space in terms of its exterior Hausdorff measure as follows (2). Let (δ, X) be any enumerable class of sets whose point-set union contains X and whose members are each of diameter less than δ. Let (δ, X) denote the class of all (δ, X).

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show that

where jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the form

and the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).

In a recent paper (1) I studied a class of generalized convex functions of a single real variable which I called sub-(L) functions. Given an ordinary linear differential equation of the second order L(y) = 0, a function f(x) is sub-(L) in (a, b) if it is majorized there by the solutions of the equation. More precisely, for every x1, x2 in (a, b),f(x) ≤ F12(x) in (x1, x2), where F12 is that solution of L(y) = 0 (supposed unique) which takes the values f(xi) at xi. It was found that sub-(L) functions are characterized in a manner closely analogous to ordinary convex functions.

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussage

kann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.

In connexion with moment problems, S. Verblunsky proved the following two theorems:

Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,

This sequence of papers is concerned with a group of theorems on Fourier transforms, and with the corresponding results for Fourier series. The transform theorems deal with the equations

when two of the functions concerned are monotonic; and the series theorems, with the equations

when two of the functions or sequences of coefficients are monotonic. A general survey of the problem was given in two earlier papers, which we shall call M.F. (I) and M.F. (II), and the results obtained were tabulated in M.F. (II). There is thus no need to restate the position in detail here. We shall merely recall as briefly as possible what has been proved already, so as to show the scope of the present paper.

Several recent papers have been concerned with a group G, of order 28.36.5.7, which can be represented as a collineation group in five dimensions. This collineation group is generated by harmonic inversions (projections) which leave fixed a point (the vertex) and a prime (said to be conjugate to the vertex). There are 126 projections in G and the set of 126 vertices form a configuration which is described in a paper (Hamill (3)) in which the operations of G are expressed as products of at most six projections. The group leaves invariant six algebraically independent primals; these have been determined (Todd (6)), and the equations of the simplest are given. The simplest invariant is a sextic primal, and some properties of this have been recorded in an earlier paper (Hartley (4)).