86 
MATHEMATICS: HARDY AND LI TTLEWOOD 
Weyl's ^principle' enables him to deduce, with singular ease and ele- 
gance, theorems of 'uniform distribution' from theorems of the char- 
acter of that of §2, and to generalise them immediately to multidimen- 
sional space. It enables him to prove, for example, that the points whose 
coordinates are 
(np aq) (p = 1, 2, . . . , k; q = 1, 2, , , . , I; n = 1, 7, 3, . , .) 
where ai, 0:2, . . . . , is any set of linearly independent irrationals^ 
are uniformly distributed in the ^unit cube^ of kl dimensions. All that we 
had been able to prove was that the points were everywhere dense in 
the cube. 
4. (c) Corresponding questions arise in connection with an arbitrary 
increasing sequence Xi, X2, X3, Are the points (Xn a), for 
example, uniformly distributed? The answers to such questions in 
general involve an unspecified exceptional set of values of a of measure 
zero, instead of (as when Xn = w*) a specified set such as the rationals; 
they are, in other words, only 'almost always' true. 
In our first paper in the Acta we proved quite generally that the set 
(Xn a) is almost always everywhere dense. The corresponding theorem 
of 'uniform distribution' we discussed only in one especially interesting 
particular case, that in which Xn = a", where a is an integer. The 
theorem is in this case substantially equivalent to results obtained by 
Borel,^ from the standpoint of the theory of probabilities, and by Faber,^ 
as a corollary of Lebesgue's theorem that a rectifiable curve has a tan- 
gent at almost every point. Our analysis however contains the first 
direct and general discussion of the problem, and leads to results nota- 
bly more precise than that of mere uniformity of distribution. These 
results were, afterwards made the subject of important generalisations 
by Fowler,2 whose investigations covers all cases in which Xn increases 
with tolerable regularity and as fast as an exponential of the type e»^. 
Weyl's 'principle' enables him to reduce this problem to a study of the 
series (Xn a), and leads him to the following theorem, so far the most 
general of its kind. If c > 0, d > 0, and Xn increases by at least c 
whenever n increases from h by as much as h (log h)~^~^, then 
and the points (Xn a) are uniformly distributed, for almost all values of a. 
In our second paper in the Acta we stated that the equation could, 
in very many cases, be replaced by the much more precise equation 
n 
(1) 
j„ = 0 
(2) 
