84 
MATHEMATICS: HARDY AND LI TTLEWOOD 
SOME PROBLEMS OF DIOPHANTINE APPROXIMATION: 
THE SERIES 2e(Xn) AND THE DISTRIBUTION OF 
THE POINTS {\na) 
By G. H. Hardy and J. E. Littlewood 
TRINITY COLLEGE. CAMBRIDGE. ENGLAND 
Commuaicated by E. H. Moore, December 5. 1916 
1. In our previous writings on the subject of Diophantine approx- 
imation, which we refer to in a short note published in the October 
number of these Proceedings/ we alluded in several places to a series 
of further results which, we hoped, were to form the material for a third 
memoir in the Acta Mathematica. The prosecution of this work was 
delayed, in the first instance, by our occupation on a long 'memoir on 
the theory of the Riemann Zeta-function, now in type and shortly to 
appear there, and subsequently by other causes; and there is, under 
present conditions, little hope of its completion in the immediate future. 
The subject has since been reopened by the appearance of work by other 
writers,^ and in particular of a very beautiful memoir by Weyl in the 
latest number of the Mathematische Annalen} This paper contains 
allusions to our unpublished work: and it seems desirable that we 
should make some more definite statement than has appeared hitherto 
of our results and the relations in which they stand to WeyFs. 
The main problems which we considered were three. 
2. (a) The first problem was that of proving that, if e{x) = e^'^** 
and 
\ = an^ + n^~^ + . . .-\-ak 
is a polynomial in n with at least o ne irrational coefficient, then • 
n 
1 
We may plainly suppose that every a has been reduced to its residue 
to modulus unity: and there is no substantial loss of generality in sup- 
posing the first coefficient irrational. 
This theorem we enunciated first, in the special case in which Xn = 
an^, in our communication to the Cambridge Congress, characterising 
the proof as ^intricate.' In our second memoir in the Acta we discussed 
in detail the case k = 2, using a transcendental method which leads to 
a whole series of more precise results; and we promised a proof of the 
more general theorem in the third memoir of the series. Weyl's memoir 
contains a complete statement and proof, both quite independent of 
ours, of the theorem in its most general form. 
