MATHEMATICS: HARDY AND LITTLEWOOD 
85 
The limitation on the form of Xn, which appears in the theorem as we 
stated it, was introduced merely for the sake of compactness of expres- 
sion and does not correspond to any real simplification of the problem. 
Our argument indeed depends upon an induction which compels us to 
consider the problem generally. The most comprehensive result which 
appears in our analysis is as follows: given any positive numbers e and rj 
we can determine v (e), N (e, r;), and a system of intervals j, including all 
rationals whose denominators are less than v, and of total length less than 
7), so that \ Sn\ < en for n > N, all values of a exterior to the intervals j, 
and all values of ai, ^2, . . • . , a*. From this result it follows at 
once that Sn = o (n) for any particular irrational a, and uniformly in 
ai, Q!2, . . . . , a*. 
Weyl's proof and ours are widely dilfferent, and each, we hope, may 
prove to have an interest of its own. The same is true of the deduc- 
tion of the formula ^ {I -\- it) = o (log t), made by Weyl as well as by 
ourselves. 
3. (b) The second principal problem was, to use Weyl's phraseology, 
that of the 'uniform distribution' (Gleichverteilung) of the points (Xn) 
where (x) is the residue of x to modulus unity. Suppose that m is the 
number of the first n such points which fall within an interval J of 
length 8. Then the points are said to be uniformly distributed if nj dn for 
every such interval j. It is plain that a corresponding definition may 
be given of uniform distribution of an enumerable sequence of points 
in space of any number of dimensions. 
That the points (X„) are uniformly distributed when k = I and a is 
irrational was proved independently by Bohl, Sierpinski, and Weyl 
in 1909-10. The general result (with the same unessential limitation 
as to the form of Xn) was stated by us in our first paper in the Acta. 
Our proof, which has never been published, proceeded on the same lines 
as that of the thoerem of §2. But Weyl has now established a 'prin- 
ciple' which renders such a proof entirely unnecessary, and which has 
led him to results in this direction far more comprehensive than any of 
ours. This 'principle' is expressed by the theorem: if 
^e(m\k) = o(n) 
1 
for every positive integral value of m, then the points (Xn) are uniformly 
distributed in (0, 1). The proof depends on a simple but ingenious use 
of the theory of approximation to arbitrary functions by finite trigo- 
nometrical polynomials; and there is a straightforward generalisation 
to space of any number of dimensions. 
