SOME PROBLEMS OF DIOPHANTINE APPROXIMATION.
Bx
G. H. HARDY and J. E. LITTLEWOOD,
TRINITY COLLEGE~ CAMBRIDGE.
I . 00.
so t h a t
I~
T h e f r a c t i o n a l p a r t o f nkO.
z . o - - I n t r o d u c t i o n .
L e t us denote by [z] a n d (x) the integral and fractional parts of x,
0 : ) - - x - - [~:], o <_ (x) < I .
L e t 0 be an irrational number, a n d a a n y number such t h a t o < a < x . Then it is well known t h a t it is possible to find a sequence of positive integers nl, n2, n3,.-- such t h a t
( I . OOI) ( n r O) -"* 6~
as r ~ o .
I t is necessary to insert a few words of explanation as to the meaning to be a t t r i b u t e d to relations such as ( I . ooi), here a n d elsewhere in the paper, in the particular case in which a = o. The formula ( i . ocT), when a > o, asserts t h a t , given a n y positive n u m b e r ~, we can find r0 so t h a t
- - e < (n~ 0 ) - - a < ~ (r > r0).
The points (n~ O) m a y lie on either side of a. B u t (n~ O) is never negative, and so, in the particular case in which a = o, the formula, if interpreted in the obvious manner, asserts m o r e t h a n this, viz. t h a t
o_< (m. O) < e (r > r~
156 G. H. Hardy and J, E. Littlewood.
The obvious i n t e r p r e t a t i o n therefore gives rise to a distinction between the value a = o a n d other values of a which would be exceedingly inconvenient in our subsequent analysis.
These difficulties m a y be avoided by agreeing t h a t , when a = o, the formula ( I . ooi) is to be interpreted as meaning r set o/ points (n~ O) has, as its sole limiting point or points, one or both o/ the points 1 and 0', t h a t is to say as imp-
ying t h a t , for a n y r greater t h a n r0, one or other of the inequalities o < ( n , 0 ) < , , z - - , < ( n ~ O ) < z
is satisfied. In the particular case alluded to above, this question of interpreta- tion happens to be of no importance: our assertion is true on either inter- pretation. B u t in some of our later theorems the distinction is of vital im- portance.
Now let ] (n) denote a positive increasing function of n, integral when n is integral, such as
n , n z , n s , . . . 2 n , 3 n , . . - , n ! , 2 n z , ' . - 2 z n , - ' ' .
The result stated at the beginning suggests the following question, which seems to be of considerable interest: - - For what /orms o/ / ( n ) is it true that, /or any irrational O, and any value o/ a such that o ~ a < I, a sequence (n~) can be /ound such that
(~.
002)
(l (n,.) O ) - a ?I t is easy to see t h a t when the increase of /(n) is sufficiently rapid the result suggested will not always be true. Thus if l ( n ) = 2 " a n d 0 is a n u m b e r which, expressed in the b i n a r y scale, shows at least k o's fo]lowing upon every I, it is plain t h a t
(2"0) < ~- + ilk, 2
when ;,k is a number which can be made as small as we please by increasing k sufficiently. There is thus an ~)excluded interval~ of values of a, the length of which can be made as near to 89 as we please.
If/(n)=
3" we can obtain an ex- cluded interval whose length is as near to { as we p]ease, a n d so on; while if ](n) = n! it is (as is well known) possible to choose 0 so t h a t (n!0) tends to a unique limit. Thus (n!e)-- o.At the end of the paper we shall r e t u r n to the general problem. The im- mediate object with which this paper was begun, however, was to determine whe-
Some problems of Diophantine Approximation. 157
ther the relation (i .oo2) always holds (if 0 is irrational) when ](n) is a power of n, and we shall be for the most part concerned with this special form of ] (n).
i . oi. The following generalisation of the theorem expressed by (I. ooi) was first proved by KaOI~ECKER2
Theorem 1 . 0 1 . relation el the type
I / 0~, 0 2 , " " Om are linearly independent irrationals (i. e. i / n o
al 01 + a2 0~ + . . . + am O,n + am+ 1 ~ o ,
where a,,a2,...a,,,+~ are integers, not all zero, holds between 0~, 02,'.'0,,~), a n d ~ , a2, " ' ~ , , , are numbers such that o < % < i , then a sequence (n~) can b e / o u n d such that
(nrOl) -~ al, (nr 02) -~ a2, " " , (n~ 0,,) -* a,,,
as r ~ ~ . Further, in the special case when all the a's are zero, it is unnecessary to make a n y restrictive hypothesis concerning the O's, or even to suppose them irrational.
T h i s t h e o r e m a t o n c e s u g g e s t s t h a t t h e s o l u t i o n o f t h e p r o b l e m s t a t e d a t t h e e n d o f I . o o m a y b e g e n e r a l i s e d a s f o l l o w s .
T h e o r e m 1 . 0 1 1 . I / Or, 0~, ... 8m are linearly independent irrationals, and the a's are a n y numbers such that o < a < I , then a sequence (n~) can be / o u n d such that
(I. OII)
(n,01) a.~t, (n, O2)-~a2~,'",(n,O,~) a2,~,I
n k1 KROI~ECKER, Berliner Sit~ungsberichte, 11 Dec. 1884; Werke, vol. 3, p. 49.
A n u m b e r of special cases of t h e t h e o r e m w e r e k n o w n before. T h a t in w h i c h all t h e a's are zero was g i v e n by DIRICHLET (Berllner Sitzungsbe~qchte, 14 .April 1842, Werke, vol. 1, p. 635). W h o first stated e x p l i c i t l y t h e special t h e o r e m s in w h i c h m = I we h a v e b e e n unable to discover. DIRICHLET (1. C.) r e f e r s to t h e s i m p l e s t as ~li~ngst bekannt* : it is of course an i m m e d i a t e c o n s e q u e n c e of t h e e l e m e n t a r y t h e o r y of s i m p l e c o n t i n u e d fractions. See also MII~KOWSKI,
~Diolahantisehe Ap2aroximation*, pp. 2, 7. KROI~ECKER'S g e n e r a l t h e o r e m has been r e d i s c o v e r e d i n d e p e n d e n t l y by several writers. See e. g. BOREL, Lemons sur les sdries divergentes, p. 135; F. Rmsz, Comptes l~endus, 29 Aug. 1904. Some of t h e ideas of w h i c h we make m o s t use are v e r y similar to those of t h e l a t t e r paper. I t should be added t h a t DIRICHLET'S and KaOI(~.CKER'S t h e o r e m s are p r e s e n t e d by t h e m m e r e l y as p a r t i c u l a r cases of m o r e g e n e r a l t h e o r e m s , w h i c h h o w e v e r r e p r e s e n t e x t e n s i o n s of t h e t h e o r y in a d i r e c t i o n d i f f e r e n t f r o m t h a t w i t h w h i c h we are con- Cerned.
A n u m b e r of v e r y beautiful applications of KRO:NECKER'S th e o r e m to t h e t h e o r y of t h e RIE~X~ c-function h a v e b e e n m a d e by H. BOHR.
158 G.H. Hardy and J. E. Littlewood.
Further, i] the a's are all zero, it is unnecessary to 8 u ~ o s e the O'a restricted in any way.
i . 02. This theorem is the principal result of the paper: it is proved in section x. 2. The remainder of the paper falls into three parts. The first of these (section x. x) consists of a discussion and proof of KRO~.CK~.R'S theorem.
We have thought it worth while to devote some space to this for two reasons.
In the first place our proof of theorem 1. 011 proceeds b y induction from k to k + x, and it seems desirable for the sake of completeness to give some account of the methods b y which the theorem is established in the case k ~ I. In the second place the theorem for this case possesses an interest and importance suffic- ient to justify any a t t e m p t to throw new light upon it; and the ideas involved in the various proofs which we shall discuss are such as are important in the further developments of the theory. We believe, moreover, t h a t the proof we give is considerably simpler than a n y hitherto published.
The second of the remaining parts of the paper (section x. 3) is d e v o t e d to the question of the rapidity with which the numbers (n~O~) in the scheme (x. oix) tend to their respective limits. Our discussion of the problems of this section is very tentative, and the results v e r y incomplete; 1 and something of the same kind m a y be felt a b o u t the paper as a whole. We have not solved the problems which we a t t a c k in this paper with anything like the definiteness with which we solve those to which our second paper is devoted. The fact is, however, t h a t the first paper deals with questions which, in spite of their more elementary appearance, are in reality far more difficult than those of the second. Finally, the last section ( I . 4) contains some results the investigation of which was sug- gested to us b y an interesting theorem proved b y F. BERNSTmN. s The disting- uishing features of these results are t h a t t h e y are concerned with a single irrat- ional 0 and with sequences which are not of the form (nkO), and t h a t t h e y hold for almost all values of 0, i. e. for all values except those which belong to an exceptional and unspecified set of measure zero.
. ~ - - K r o n e e k e r ' s T h e o r e m .
i . IO. KRONECKER'S theorem falls naturally into two cases, according as to whether or not all the a's are zero. We begin b y considering the simpler case,
1 Some of t h e results t h a t we do obtain, h o w e v e r , are i m p o r t a n t f r o m t h e p o i n t of v i e w of applications to t h e t h e o r y of t h e series ,~ ent~0i and t h a t of t h e RI~.~XNS ~-function. I t was in p a r t t h e possibility of t h e s e applications t h a t led us to t h e researches w h o s e results are g i v e n in t h e p r e s e n t paper. T h e applications t h e m s e l v e s will, we hope, be g i v e n in a l a t e r paper.
Math. Annalen, vol. 71, p. 421.
Some problems ot Di0phantine Approximation. 159 when all the a's are zero. Unlike most of the theorems with which we are con- cerned, this is not proved b y induction, and there is practically no difference between the cases of one and of several variables. The proof given is :DIRICH- LET'S.
Let ~ denote the number which differs from x b y an integer and which is such t h a t - - 8 9 < ~ < 8 9 Then the theorem to be p r o v e d i s equivalent to the theo- rem that, given a n y integers q and N, we can find an n not less than N and such t h a t
InO, l < zlq, InO, l< zlq, ..', In#,,, I < ~lq.
L e t us first suppose t h a t h r -~ T. Let R be the region in m-dimensional space for which each coordinate ranges from o to z. L e t the range of each coordinate be divided into q equal parts: R is then divided into qm parts. Consider now the ~ + T points
0'0,), (~,0~), ..., (~,0~); (~,---- o, z, 2, ...q").
There must be one p a r t of R 'which contains two points; let the corresponding values of v be % and v2. Then clearly
I (~, - ~,) o, 15 z/q, I (~, - ~,) o, I <_
T/q,...,
I(~, -- ",) 0,,, I <__T/q,
a n d 1% - - v ~ I --> z.
We have therefor only to take n - - - [ % - - v ~ [. We observe t h a t we have also n < q ' ,
a result to which we shall have occasion to return in section i . 3.
If N > z we have only to consider the points ( v N 0 , ) , (vN0~),-.. instead of the p o i n t s (v01), (v02),.-..
i . I i . We turn now to the case when the a's are not all necessarily zero.
In this case the necessity of the hypothesis t h a t the 0's are linearly independent is obvious, for the existence of a linear relation between the 0's would plainly involve t h a t of a corresponding relation between the a's; naturally, also, the added restriction makes the theorem much more difficult than the one just proved.
Our proof proceeds b y induction from m to m + i; it is therefore i m p o r t a n t to discuss the case m---i. The result for this case m a y be proved in a variety of ways, of which we select four which seem to us to be w o r t h y of separate dis-
160 G . H . Hardy and J. E. Littlewood.
cussion. These proofs are all simple, and each has special advantages of its own.
It is i m p o r t a n t for us to consider very carefully the ideas involved in them with a view to selecting those which lend themselves most readily to generalisation.
For example, it is essential t h a t our proof should make no appeal to the t h e o r y of continued fractions.
(a). The first proof is due to KROI~.CKER. I t follows from the result of i . i o , with m = i , or from the t h e o r y of continued fractions, t h a t we can find an arbitrarily large q such t h a t
and so
(~. I ~ )
where
O_--P_+ ~ q q-i'
q O - - p - ~ / q .
I,~l<~.
I t is possible to express a n y integer, and in particular the integer {q a}
nearest to q a , in the form
q n t + p n
where n and nl are integers, and
I nl<q/2.
F r o m the two equationswe obtain
and so
o r
qO--p=~]q, qn,+ "pn---{qa}
q(nO +
n t ) - - - n d +qa +
I- - i < q ( n O
+ n l - - a ) < i , l ( n O ) - - a l <~lq.
la, l<~,
If we write v = n + q a n d use ( I .
III),
w e see t h a tso t h a t
Ib'O)--.l<2/q, q/2<v<3q/2;
I (~" o) - . I < 31v
Some problems of Diophantine Approximation. 161 f o r some v a l u e of v between
q/2
a n d 3q[2. This e v i d e n t l y establishes t h e t r u t h of t h e t h e o r e m .I f we a t t e m p t to e x t e n d this p r o o f to t h e case of several variables we find n o t h i n g to c o r r e s p o n d to t h e e q u a t i o n
{qa) = grit
+ p n .
B u t KaOI~.CKER'S p r o o f has, as a g a i n s t t h e proofs we shall n o w discuss, t h e v e r y i m p o r t a n t a d v a n t a g e of f u r n i s h i n g a definite r e s u l t as to t h e order of t h e ap- p r o x i m a t i o n , a p o i n t to which we shall r e t u r n in 1 . 3 .
(b). L e t e be an a r b i t r a r y positive c o n s t a n t . B y t h e result of x . io, we c a n f i n d an n such t h a t o < 0 ~ < , or i - - e < 0 t < i , where 0t~---(n0 ). Since 0 is i r r a t i o n a l , 0~ is n o t zero. L e t us suppose t h a t o</91 < , ; t h e a r g u m e n t is sub- s t a n t i a l l y t h e same in t h e o t h e r case. W e can find a n m s u c h t h a t
mO~ < a < (m + x ) O : ,
a n d so
]toOl--a]< 01;
I ( n m O ) - - a l < ~ , w h i c h proves t h e theorem.
(c). 1 L e t S d e n o t e t h e set of p o i n t s (n0). S r, its first d e r i v e d set, is closed.
I t is m o r e o v e r plain t h a t , if a is n o t a p o i n t of S r, t h e n n e i t h e r is (a + nO) n o r (a - - n 0).
The t h e o r e m to be proved is clearly e q u i v a l e n t t o t h e t h e o r e m t h a t S ~ con- sists of t h e c o n t i n u u m (%1). Suppose t h a t this l a s t t h e o r e m is false. T h e n t h e r e is a p o i n t a which is n o t a p o i n t of S r, a n d therefore a n i n t e r v a l con- t a i n i n g a a n d c o n t a i n i n g ~ no p o i n t of S t. Consider I, t h e greatest possible s u c h i n t e r v a l c o n t a i n i n g a. a T h e i n t e r v a l o b t a i n e d b y t r a n s l a t i n g I t h r o u g h a d i s t a n c e
O,
a n y n u m b e r of times in either direction, ~ m u s t , b y w h a t was said above, also c o n t a i n no p o i n t of S t. B u t t h e i n t e r v a l t h u s o b t a i n e d c a n n o t over- lap w i t h I, for t h e n I would n o t be t h e ,)greatest possible, i n t e r v a l of its kind.This proof was discovered independently by F. Rmsz, but, so far as we know, has not been published.
s In its interior, in the strict sense.
B The existence of such a ~greatest possible~ interval is easily established by the classical argument of DEDEKIND.
4 Taking the congruent interval in
(o, I).
This interval may possibly consist of two separ- ate portions (o, $1), and ($~, i).Acta mathematica. 37. Imprim6 le 9-5 f~vrier 1914. 21
162 G . H . Hardy and J. E. Littlewood.
Hence, if we consider a series of [i/d] translations, where d is the length of I, it is clear t h a t two of the corresponding [I/J] + I intervals must coincide.
Clearly this can only h a p p e n if 0 is rational, which is contrary to our hypothesis.
(d). We argue as before that, if the theorem is false, there is an interval I, of length 2 ~ and middle point a, containing no point of ~'. B y the result of I . io we can find n so that, if fl~-~ (nO), then o < 01 < ~ or I - - ~ ~ 0~ < z.
B y the reasoning used in (c) it appears t h a t the interval obtained b y trans- lating I through a distance 01, a n y number of times in either direction, must contain no point of ft. But since each new interval overlaps with the preceding one it is clear t h a t after a certain n u m b e r of translations we shall have covered the whole interval o to I b y intervals containing no point of ~q', and shall thus have arrived at a contradiction.
I . I2. Let us compare the three last proofs. I t is clear t h a t (b) is consid- erably the simplest, and t h a t (d) appears to contain the essential idea of (b) together with added difficulties of its own. I t appears also that, in point of simplicity, there is not v e r y much to choose between ( c ) a n d (d), and t h a t (c)has a theoretical a d v a n t a g e over (d) in t h a t it dispenses the assumption of the theorem for the case a ~ o, an assumption which is made not only in (b) and (d), b u t also in (a). When, however, we consider the theorem for several variables, it seems t h a t (b) does not lend itself to direct extension at all, t h a t the com- plexity of the region corresponding to I in (c) leads to serious difficulties, and t h a t (d) provides the simplest line of argument. I t is accordingly this line of argument which we shall follow in our discussion of the general ease of KRO~- ECKER'S theorem.
I . 13. We pass now to the general case of KRO~ECKER'S theorem. We shall give a proof b y induction. F o r the sake of simplicity of exposition we shall deduce the theorems for three independent irrationals 0, 9o, ~ , from t h a t for two. I t will be obvious that the same proof gives the general induction from n to n + I irrationals.
We wish to show that if we form the set ~ of points within the cube o < ~ < I , o < y < I , o < z < I, which are congruent with
(0, ~0, ~v), (20,
2q~,
2tp), 9 ...(nO, n~o,
n~P), -..then every point of the cube is a point of the first derived set 8'. I t is plain that, if (a, #, ~,) is not a point of 8, then neither is ((a +
nO), (~ + n~o), O' +ntp))
nor((a--nO), (fi--nqo), (r--nip)).
If now our theorem is not true, there m u s t exist a sphere, of centre (~, ~, 7) and radius Q, which contains t no point of ~'. B yWithin or upon the boundary.
Some problems of Diophantins Approximation. 163 the resuIt of I . Io, there is an n such t h a t the distance c~ of ((nO), (nr (n~p)) or (01, r ~0~) from one of the vertices of the cube is less than ~/1/2-2. Let us sup- pose, for example, t h a t the vertex in question is the point (o, o, o). Consider t h e straight line
(I
.I3I )
g~--6r y - - ~ _ _ Z - - 7" T
a n d t h e infinite cylinder of radius $ with this line as axis. It is clear t h a t t h e finite cylinder C obtained by taking a length $ on either side of (a, fl, 7 ) i s en- tirely contained in the sphere and therefore contains no point of SL Hence t h e cylinder obtained by translating G through (0~, ~ , ~p~), a n y n u m b e r of times in either direction, also contains no point of S t, so that, since each new position of G overlaps with the preceding, the whole of the infinite cylinder, or r a t h e r of the congruent portions of the cube, is free from points of S ~.
Let us now consider the intersections of the t o t a l i t y of straight lines in t h e cube, which are congruent with portions of the axis of the cylinder, with an a r b i t r a r y plane 9 ~ x 0. We shall show t h a t t h e y are e v e r y w h e r e dense in the square in which the plane cuts the cube, whence clearly follows t h a t no point of the cube is a point of St, and so a contradiction which establishes the theorem.
The intersections (y, z) are congruent with t h e intersections of the axis ( I . ISI ) w i t h
X=~O"['- V, (~] . . . . , - - 2 , - - ' r 0, I , ~ , ' ' ' ) ,
a n d so t h e y are the p o i n t s congruent with
(Xo--a)r + ~'r (xo--a)~Pl "~Pl
fl + 01 0~ ' 7 + 01 + 0--(-
But, under our hypothesis, cpl/0~ and ~01/01 are linearly i n d e p e n d e n t irrationals, a n d so, b y the theorem for two irrationals, this set of points is e v e r y w h e r e dense in the square. The proof is thus completed.
x . i 4. We add two f u r t h e r remarks on t h e subject of KRONECKER'S t h e o - rem, in which, for t h e sake of simplicity of statement, we confine ourselves to the case of two linearly independent irrationals 0, ~.
(a) Suppose t h a t o < a < I, o < fl < I. KRON~CKER'S theorem asserts the exis- tence o~ a sequence (n,) such t h a t (ne0)---a, (ns~)--*fl. L e t us choose a se- quence of points
164 G. It. Hardy and J. E. Litflewoed.
s u c h t h a t
T h e r e is, for a n y v a l u e of ?t, a s e q u e n c e {n~ s u c h t h a t
(n.,O)
- .a., (n..
~ ) - . ~ . ,as s - . oo. F r o m t h i s it is e a s y t o d e d u c e t h e e x i s t e n c e of a s e q u e n c e (n~) f o r which (n~O) a n d (n~ep) t e n d t o t h e limits a a n d fl a n d are a l w a y s g r e a t e r t h a n t h o s e limits, so t h a t t h e d i r e c t i o n of a p p r o a c h t o t h e limit is in each case f r o m t h e r i g h t h a n d side. 1 Similarly, of course, we can e s t a b l i s h t h e e x i s t e n c e of a s e q u e n c e giving, f o r e i t h e r O o r r e i t h e r a r i g h t - h a n d e d or a l e f t - h a n d e d a p - p r o a c h t o t h e limit.
I f we a p p l y similar r e a s o n i n g t o t h e case in w h i c h a o r fl or b o t h a r e z e r o we see t h a t , w h e n 0 a n d ~ a r e l i n e a r l y i n d e p e n d e n t i r r a t i o n a l s , we m a y a b a n d o n t h e c o n v e n t i o n w i t h r e s p e c t t o t h e p a r t i c u l a r v a l u e o w h i c h was a d o p t e d in i . oo, a n d a s s e r t t h a t t h e r e is a s e q u e n c e for w h i c h (nO)---,a a n d ( n ~ ) - - ~ , a a n d fl h a v i n g a n y v a l u e s b e t w e e n o a n d I, b o t h v a l u e s i n c l u d e d , a n d t h e f o r m u l a e h a v i n g t h e o r d i n a r y i n t e r p r e t a t i o n . T h i s r e s u l t is t o b e c a r e f u l l y d i s t i n g u i s h e d f r o m t h a t of i . io. T h e l a t t e r is, t h e f o r m e r is not, t r u e w i t h o u t r e s t r i c t i o n o n t h e O's, as m a y be seen a t o n c e b y c o n s i d e r i n g t h e case in w h i c h ep = - O.
(b). I t is e a s y t o d e d u c e f r o m KRONECK~.R'S t h e o r e m a f u r t h e r t h e o r e m , w h i c h m a y be s t a t e d as follows: I if we talce any portion 7 of the square o < x <_i, o < y < i , bounded by a /inite number o/regular curves, and of area ~ ; and if we denote by N r (n) the number o/ the points
((~0), (~r (~
= I , z , . . . n ) ,which /all inside 7; then
a 8 n . - - , o o .
T h i s result, w h e n c o m p a r e d with t h e v a r i o u s t h e o r e m s of this p a p e r , sug- gests a w h o l e series of f u r t h e r t h e o r e m s . T h e p r o o f s of t h e s e a p p e a r l i k e l y t o b e v e r y difficult, a n d we h a v e , u p t o t h e p r e s e n t , c o n s i d e r e d o n l y t h e case of a single i r r a t i o n a l 8. W e h a v e p r o v e d t h a t , if N~ (n) d e n o t e s t h e n u m b e r of t h e p o i n t s
(~" 0), (~
= i , 2 , . . . n ) ,I The reasoning by which this is established is essentially the same as that of I. 20.
This is a known theorem. For a proof and references see the tract 'The Riemann Zeta-function and the Theory of Prime Numbers', by H. Bo~z and 3". E. LITI~.EWOOD, shortly to be published in the Cambridge Tracts in Mathematics and Mathematical Physics.
Some problems of Diophantine Approximation. 165 which fall inside a segment 7 of (o, x), of l e n g t h ~, then N r (n)c~ ~n. This re- sult m a y be compared with t h a t of Theorem 1 . 483 at the end of the paper.
B u t results of this character will find a more natural place among our later investigations t h a n among those of which we are now giving an account.
~. 2. - - T h e g e n e r a l i s a t i o n o f K r o n e e k e r ' s t h e o r e m .
i . 20. We proceed now to the proof of theorem 1. 011. Our a r g u m e n t is based on the following general principle, which results from the work of Pm~Qs- HEI~ a n d LO~DO~ o n double sequences a n d series?
1 . 2 0 . I1
lira lira ... lira Ip (r,, r2, ... rD = A , , (p---- i, 2, ... m),
rl--~ oo r~ --~ o9 r k ---~oo
then we can l i n d a sequence o / s e t s (rl~, r 2 n , ' . , rk,~) such that, as n ~ oo,
a n d
icular k, then it is true for ]r x.
t h u s have proved it generally.
We shall abbreviate 'lira lira
r I - - - ~ r 2 ----~
no danger of confusion, into 'lira'.
L e t
lim l~, (r,, r , ,
r , , r 2 , . . r~
Then by hypothesis
a s r k + l -'-* o o .
r q , - - | ( q = I, 2 , . . . k),
tp (rl,, r~,, ... rk,)-~ Ap, (p = I, 2 , . . . m).
We shall show that, if this principle is true for all values of m and a part- As it is plainly true for k---x, we shall 9 .- l i m ' into 'lira' , or, when there is
r k - , o o r I , r~ ,- 9 r k
9 .- r~+l) = lp (rk+l).
B y the principle for b variables, we can find rx~, r~., 2 n, a n d such t h a t
lp (rk+l) -* Ap
L e t us choose an integer rk+l,n, greater t h a n 2 n, for which I/~ (rk+l,,) - - Ap] < 2 - - - 1 , (p = ~, a , . - . m).
9 -. rk~, all greater t h a n
i PalNGStIEIM, MUnchener Sitzungsberichte, e e l . 27, p. 101, a n d Math. Annalen, eel. 53, p. 289;
LONDON, Math. Annalen, eel. 53, p. 322.
166 G . H . Hardy mad J. E. Littlewood.
I/P (rln, r 2 n , ' ' ' rkn,, rk+l,n) - - / p (rk+l,.)l < g--n---l, (p ~_. I, 2 , ' ' ' m).
We thus obtain a sequence of sets (r~., r 2 . , ' . , r~+~,.), such t h a t every member of the n ta set is greater t h a n 2" and
lip(r,,,, r2.,-.- r k + , , . ) - - A p [ < z - " , ( p = I , 2 , ' ' " m).
This sequence evidently gives us what we want.
An i m p o r t a n t special case of the principle is the following:
1. 201. I ] /or all values of t we can ]ind a sequence n i t , nst,. ., n~t, ... such that
[p(nrt) ~ A p t , ( p = I, 2 , ' ' " ~'i,), as r--. | and i/
Apt--*Ap, (p = I, 2 , . . . m ) , as t - - ~ , then there is a sequence (ns) such that
/ p ( n o ) - - . A p , (p = ~, 2 , . . . m),
a8 8 - - - . aO .
This is in reality merely a case of the principle t h a t a limiting-point of limiting-points is a limiting-point.
I . 2L We consider first the case in which all the a's are zero, and the O's are unrestricted. In this case the proof is c o m p a r a t i v e l y simple.
T h e o r e m 1 . 2 1 . There is a sequence ( n , ) such that, as r---. oo (n~ Op) - - O, ( X = Is 2,''" ]~; p = I, 2 , - " m).
We prove this theorem b y induction from k to b + I: we have seen t h a t it is true when b----x. We suppose then t h a t there is a sequence (~,) such t h a t (I 9 2II) (#," Op) --- o, ( x = i , 2 , . . . k ; p = i , 2,...ra).
The sequence
uk+10 ~ /uk+10 ~ tu~+10 ~ ( s = I, 2,---),
has at least one limiting point ~,, ~2,..-cpm; hence, b y restricting ourselves to a subsequence selected from the sequence (p,), we can obtain a sequence (~9 such that, as s - - ~ ,
O,:op)--o, (x_< k); (v~+'op)--~p;
( p = ~, ~,...m).Some problems of Diophantine Approximation. ~67 We then have, for x < k + I,
where the G's are constants, zj-[-~2 § 2 4 7 and uq<]r In virtue of ( I . 2 I i ) we can evaluate a t once e v e r y repeated limit on the right hand side, and it is clear t h a t we o b t a i n i t , s or o according as u----k + i or u~]c. I t follows from the general principle 1.9.0 t h a t we can find a sequence (nr~), (r ~ I, 2,...), such that, as r--~ ~ ,
k + l __.
(n~Op)--.o, ( ~ k ) ; (n~ 0,) ~p; (p~-I, 2,..'m).
But, b y theorem 1 . 0 1 , we can find a sequence (its) such t h a t
(~,~)-.o, (p.-~i, 2,...m);
and we have only to a p p l y the principle 1 . 201 to obtain the theorem for k + I.
I . 22. We pass now to the general case when the a's are not all zero. We have to prove t h a t i/ fl~, Oz,..-0~ are linearly independent irrationals, there is a sequence (nr) such that, as r-.-, o0,
(n70~)--a~, (z-~i, 2,-.. k; p = I , 2,...m).
We shall p r o v e this b y an induction from /c to k + i which proceeds b y two steps.
(i). We assume the existence, for a particular k, any number m of 0's, and a n y corresponding system of a's, of a sequence giving the scheme of limits
1
n ~ t l
n ~ ~21
n k ~ k l
0 5 9 9 ~ 9
= l
a l 2 . . . . ~1 nt
/ X 2 2 . . . . / ~ 2 m
9 , ~ . ~ ~
9 9 9 ~
9 . . ~ ~ ~
O k 2 . . . . f f k n t
and we prove the existence, for any number m of 0's, and a n y corresponding system of a's, of a sequence giving the scheme
168 G. H. Hardy and J. E. Littlewood.
9t t
n k Bk+l
01 0~ . . . . 0,,,
~ 1 1 ~ 1 2 . . . . U l m
~:z ~22 . . . . q2m
9 9 , , .
9 9 , , .
~ k l ~k2 . . . . ~km
0 0 . . . . 0 9
I t will be understood t h a t neither m , nor the O's, nor the a's are necessarily the same in these two schemes, all of them being arbitrary 9
(ii). We then show t h a t we can pass from the last written scheme of limits to the general scheme in which the elements of the last row also are a r b i t r a r y . z . 23. Proo/ o/ the /irst step. To fix our ideas we shall show t h a t we can pass from a sequence (no) giving 1
n , . O - - a , , n, cp---.fl~, n~qJ--'7~, n , . x - - J , , n ~ v - - - . ~ , n , . v - - ~ ,
to a sequence (m,) giving
m~O--.al, m,q~--" ~l, m~ O - - a2 , ,nJ ~p . - ~, , m ' O . - * o , m'~p.-.o.
I t will be clear t h a t the argument is in reality of a perfectly general type.
Suppose we are given a'~, at2, ~/,, {~r2, and t h a t O, ep, at,, arz, fir,, ~ , , are ]ine- arly independent irrationals. Then b y hypothesis we can find a sequence giving t h e scheme
n,. 0 --. s nrep --.* ~'1, n,. a', --'* o, n,. ~'~ - ~ o, n, a'~ --* o, n, #' 2 - - - o, t,l O - - at,, ~ g r fir,, n~ a', - - o, n~ ~', - - o.
Further, the set of points (~'0, n~'?) has at least one limiting-point (~, ~), and, by restricting ourselves to a subsequence of (nr), we may suppose that we have also
n'O-...~., n',.ep.--t~.
I n w h a t f o l l o w s we s h a l l o m i t th• b r a c k e t s in ( n O , . . . ; it is of c o u r s e to b e u n d e r - s t o o d t h a t i n t e g e r s a r e to be i g n o r e d .
Some problems of Diophantine Approximation. 169 We express all this b y saying that we can find a sequence (nr) giving the scheme
~ t l ~ f l J t , O , O~ O~ O ,
( I . 2 2 I ) C~I2, /~t , O , O ,
4, #.
The sequence (kr), where k r = 2 n , , gives us the scheme [ 2c~tl, 2flit, O, O, O, O,
( I . 222) 4a'2, 4fl'2, o, o,
8 4, 8it.
B y the generM principle 1 . 2 0 , we can find a sequence
(lr)
giving the scheme lim (n~, + n~z + " - + n~,) 0, lira (n~, + n~2 + . - - + nr,) go, . . . .lira (n~, + n~2 + . ' . + n~,)~ O . . . lira (m, + n,~ + . . . + n~,)~ 0 . . . .
where 'lira' stands for lira
rt, r2, 9 9 9 r8
Consider the repeated limit
lira (n~ + n~2 + - . . + nn) 38,
f'l, ~'2, " 9 "#'$
which is easily evaluated with the aid of the table ( I . 221). The limit of a term n$iO is it: t h a t of a 'cross-term'
n a n ~ n~kO ( a + b + c = 3 ; a , b , c < 3 ; i < ] < k ) r i r j
is zero, s i n c e n ~ 0 tends to an d or a flr, a n d n b d and n b f tend to zero. Thus r k r i r j we obtain the repeated limit 8it. In all the other repeated limits the cross-terms give zero in the same way, and we see t h a t the sequence (l~) gives the scheme
8at1, 8fit1, o, o, o, o, 8at2, 8fir:, o, o, 82 , 8 ~ . Oonsider now the r e p e a t e d limits
A c t a rnath~*atica. 37. Imprim6 le 27 f6vrier 1914. 22
170 G . H . Hardy and J. E. Littlewood.
lira (l~ + k~, + k r , + . - . + k~,,,) ~ r, r l , " " ' r f ~
where
X = O , ~ , a ' t , f t , d * , f ~ ; x = i , 2 , 3 .
A]I the cross-terms contribute zero as before, and we obtain the scheme (8 + 2 r n ) d ~ , (8 + 2 ~ n ) f t , o, o, o, o,
(8 + 4ra) a'2, (8 + 4m)/~'2, o, o, ( 8 + 8 m ) Z , ( 8 + 8 r a ) ~ ,
or
6a't + (ra + i)2a'1, 6~'1 + (m + I ) 2 5 ' t , o, o, o, o, 4a'2 + (m + I ) 4 a ' 2, 4fl'2 + (ra + z)4fl'2, o, o,
( m + ~ ) S Z , ( m + z ) 8!t.
I t is possible, then, to find a sequence giving this scheme. B u t now, since it is possible to find a sequence of m's such t h a t
(m + I) tp--* o, (~) m - z a ' t , 2/~'t, 4a'2, 4fl'2, 8~,, 8?L),
it follows (in virtue of the principle 1 . 2 0 ) t h a t we can find a sequence giving the scheme
6 a ' t, 6 f l ' , , o , o, o, o, 4a'2, 4fl'2, o, o,
0 ~ O.
This gives us what we want (and something more) provided it is possible to choose
I I = I ~'2 ~ ~ ' '
a', = 6 a t , fl', = ~ f l , , a', 4a~, =
This is the case provided 0, ep, ai, ~t, a~,/~2 are linearly independent irrationa]s:
it remains only to show t h a t this restriction on at, ill, a2, r2 m a y be removed.
It is obvious, in virtue of the principle 1 . 2 0 , t h a t this m a y be done provided wo can find a sequence (aim, film, a2m,/~2m) such that, for each n, 0, cp, aim,/~lm, a2~, fl2m are linearly independent irrationals, and such t h a t
N o w i t is easy to see t h a t there must be points (a~m,/~1~, a2m, f12~) interior to the 'cube' w i t h (a~, fl,, a~, fl,) as oentre and of side 2 -m, and exterior to t h a t
Some problems of Diophantine Approximation. 171 w i t h t h e s a m e c e n t r e a n d o f s i d e 2 - = - 1 , a n d s u c h t h a t 0 , 9 , a l ~ , fll~, a , ~ , fl~, a r e l i n e a r l y i n d e p e n d e n t i r r a t i o n a l s . B y s e l e c t i n g o n e s u c h p o i n t c o r r e s p o n d i n g t o e a c h v a l u e o f n w e o b t a i n a s e q u e n c e o f t h e k i n d d e s i r e d . ~
i . 24. Proo] o/ the second step. H e r e a l s o w e s h a l l c o n s i d e r a s p e c i a l c a s e f o r s i m p l i c i t y : t h e a r g u m e n t is r e a l l y g e n e r a l . W e s h a l l s h o w t h a t w e c a n p a s s f r o m a s e q u e n c e g i v i n g t h e s c h e m e
o 9 ~p z
!
n ~ a2 f12 72 ~2
n 3 0 0
t o o n e g i v i n g
0 q0
L
n Ct x ~ t
n$ ~2 ~2 n3 ~3 ~3 9
A s i n i . 23, w e m a y s u p p o s e , w i t h o u t r e a l l o s s o f g e n e r a l i t y , t h a t 0, ~f, a t , fll a r e l i n e a r l y i n d e p e n d e n t i r r a t i o n a l s . L e t ( n r ) b e a s e q u e n c e g i v i n g
0 9 a , ~1
]
ii. _I a I 2 -- ~ 3 2 f13n ~ o o 3
n 8 o o
1 This argument depends ostensibly on ZERMELO'S 'Auswahlsprinzip' (or WHITEHEAD and Russr:LL's 'Multiplicative Axiom'). This difficulty can however be surmounted with a little trouble.
I t should p e r h a p s be observed t h a t we have ignored several similar points early in the paper:
in all of these the difficulty is comparatively trivial, and we have only called attention to it in the present instance because it occurs in a more serious form t h a n is usual in constructive mathematics.
An alternative line of argument from t h a t in the t e x t proceeds as follows. I t is e a s y to show t h a t if at most a finite number of primes are omitted, any four of the sequence log2, leg 3, log ~, log 7, log i i , . . . , together with 0 and ~, form a set of six linearly i n d e p e n d e n t irrationals. ~r i t can be deduced from known results concerning the distribution of the primes that we can find a sequence (logpn, log qa, leg rn, log sn), where jgn, qn, r~, and sn are primes, such t h a t
(Iog.lon) - - ' al, (log qn) ~ ill, (log rn) -~ a,, (log s~) --, ,82.
172 Then
G. H. Hardy and J. E. Littlewood.
lim(n, +
n,)O
= lira (~ at +n,O)-~a,
lira (n~ + n , ) -~ 0 = lira(n~
a,+ n'~O)~ %,
r , S r
3 2 + n ~ O ) = a ~ ; lim
(n, + n,)SO=lim (2n, at
with similar results for ~. It follows quence giving the desired scheme, and t h a t of the theorem, is completed.
b y the principle 1 . 2 0 t h a t there is a se- the proof of the induction, and therefore
~ . 3 . - - T h e o r d e r o f t h e a p p r o x i m a t i o n .
x. 30. We have p r o v e d that under certain conditions we can find a sequence (n~) such that
( I . 3 O I )
(n~Op)-*anp
( x = I , 2 , - . . k ; p = i , 2 , . - . m ) .There are a number of interesting questions which m a y be asked with regard to the
rapidity
with which the scheme of limits is approached.The relations (I.3OI) assert that, if we are given ~t, there is a function 9 (k,m; 01,02,."0m;
au,a~2,'"ak,~; ~)1
such t h a tl
0p)I
<for some n < a}. It is hardly necessary to observe, after the explanations of i . oo, t h a t this inequality requires a modification when a ~ o = o, which m a y b e express- ed roughly b y saying t h a t a~p is then to be regarded as a two-valued symbol capable of assuming indifferently the values o and I.
(i) Does q) necessarily depend on the 0's and a's: can we for example, find a 9 independent of the a's ? It will be seen t h a t this last question is answered in the affirmative.
(ii) Can we assert anything concerning the order of q) qua function of 4, the variables 0 and a being supposed fixed? The same question m a y be asked concerning a n y 9 which is independent of the a's; it should be observed, more- over, t h a t the best answer to the latter question does not necessarily give the best answer to the former.
t For shortness we shall write this O(k, ra, O, a,~).
Some problems of Diophantine Approximation. 173 Our a t t e m p t s to answer these questions have not been successful, a n d such results as we have been able to obtain arc o f a negative character. The ques- tion then arises as to whether we can obtain more definite results by imposing restrictions on the 0's or the a's, by supposing for example t h a t all t h e d s are zero, or t h a t the 0's belong to some special class of irrationals.
(iii) The relations ( I . 3 o I ) imply the t r u t h of the following assertion: there is a function ~ ( k , m , O , a , n) which tends to i n f i n i t y with n, and is such t h a t
I(n ~ Op) - - . . p I < I / 9
for an i n f i n i t y of values of n. A series of questions m a y then be asked concern- ing 9 similar to those which we have stated with reference to O.
I . 31. We shall begin b y proving two theorems which are connected with the questions (i). The first of t h e m deals with the case in which all the a's arc zero, and it will be convenient to use in its statement, as in i . so, not the function (x), b u t the allied function ~.
T h e o r e m 1 , 3 1 . There ia a /unction q~ (k, m, Z), depending only on k, m, and
~, such that
[n"Op]<I/)~, ( x = x , 2 , . . . k ; p = i , 2 , . . . m ) , /or some n < r
F o r suppose t h a t this theorem is false. Then to every r corresponds a set of O's, say #)1, ~02,'" ~0,n, such t h a t t h e inequalities
(I. 3I~) In~m~l
< ~/zare not all true unless n > r . The set of points (~0~,,02,... ~0,,)has a t least one limiting point ( O 1 , 0 2 , " " Ore), a n d by restricting ourselves to a subsequenee of r's we can make
~o~-+o~, (p= ~,2,...m).
F r o m this it follows t h a t we can choose a number nr which tends to i n f i n i t y with r b u t so slowly t h a t
( i . 3 ~ 2 ) n ~ l ~ O p - - o p l < ~/2Z, (p = 1 , 2 , . . . m).
Clearly we m a y suppose t h a t nr<_r, a n d so we have, for an infinity of values of r, nr<r and
(I.313) nulrOp--Op[<I/2Z, (n<nr; x ~ i , 2 , - . . k ; p ~ i , 2 , . . - m ) .
174 G . H . Hardy and J. E. Littlewood.
F r o m ( i . 311) a n d ( I . 313) it follows t h a t t h e i n e q u a l i t i e s ]n~O~]<I/2),
c a n n o t all be t r u e unless n > n r , a n d so, since n r ~ o o , c a n n o t be t r u e for a n y v a l u e of n. This c o n t r a d i c t s T h e o r e m 1 . 0 1 1 .
I n t h e case k = I it is possible to a s s e r t m u c h m o r e t h a n this. I t is k n o w n , a n d is p r o v e d in I . IO, t h a t in t h i s case we m a y t a k e
( I . 314) @ = ([l] + i) 'n
This p r o b l e m , in fact, m a y be r e g a r d e d as c o m p l e t e l y solved. W h e n k > i , how- ever, t h e ease is v e r y d i f f e r e n t . W e h a v e n o t e v e n s u c c e e d e d in f i n d i n g a d e f i n - ite f u n c t i o n q)(4), t h e same for all 0's, such t h a t
In'Ol_< 1/4
for n < O . I t w o u l d be n o t u n n a t u r a l t o s u p p o s e t h a t t h e ,>best possible,>
f u n c t i o n 1 @ is less t h a n K 4, w h e r e K is a n a b s o l u t e c o n s t a n t , B u t we h a v e b e e n u n a b l e t o p r o v e t h i s or i n d e e d a n y d e f i n i t e r e s u l t as t o its o r d e r in 4.
i . 32. T h e o r e m 1 . 3 2 . I] the O's are linearly independent irrationals, it is possible to /ind a /unction @ (k, m , O , 4), independent o/ the a's, such that
I ( n ~ . o ~ ) - a ~ p l < i / 4 , ( x = I , 2 , . . . k ; p = i , 2 , . . - m ) /or some n < @.
T h a t this t h e o r e m is t r u e f o r t h e special case k = I, m--- x, follows f r o m t h e a r g u m e n t (a) in 1 . 1 1 . I t is e a s i l y p r o v e d in t h e m o s t g e n e r a l case b y a n a r g u - m e n t resembling, b u t s i m p l e r t h a n , t h a t of i . 31.
I f t h e t h e o r e m is u n t r u e , it is possible t o find a s e q u e n c e of sets (~a~p) (r ~ z, 2 , - . . ) for w h i c h t h e i n e q u a l i t i e s of t h e t h e o r e m d o n o t all h o l d unless n > r:
T h e s e q u e n c e of sets has a t l e a s t one limiting set ( ~ p ) : l e t us c h o o s e r so t h a t I - I < 1 / 2 4 , (x = I , 2 , . . . k; p ~ I , 2 , - . - m ) .
T h e n c l e a r l y t h e i n e q u a l i t i e s
I (n" I < 1/24
c a n n o t all be t r u e unless n > r , a n d so, since r is a r b i t r a r i l y large, c a n n o t all be t r u e for a n y n. This c o n t r a d i c t s T h e o r e m 1 . 0 1 1 .
That is, the function which has, for each value of k, the least possible value. For the existence of this function it is necessary that the sign ~__ above should not be replaced by <.
Some problems of Diophantine Approximation. 175 1 . 3 3 . L e t us c o n s i d e r m o r e p a r t i c u l a r l y t h e case in w h i c h k ~ I.
T h e e q u a t i o n ( I . 314) suggests t h a t it m a y in this case be possible t o choose f o r 9 a f u n c t i o n of t h e f o r m
S2 (m, 0, a) i t ' .
T h i s we believe t o be i m p r o b a b l e , b u t we h a v e n o t s u c c e e d e d , e v e n w h e n m = i , in o b t a i n i n g a d e f i n i t e proof. W h a t is c e r t a i n is t h a t no c o r r e s p o n d i n g r e s u l t is t r u e of t h e O of T h e o r e m 1 . 3 2 . I t is impossible t o choose a f u n c t i o n ~2 (m,O) i n d e p e n d e n t of it, a n d a f u n c t i o n tp (m, it) i n d e p e n d e n t of t h e 0's, in s u c h a w a y t h a t t h e q) of this t h e o r e m m a y be t a k e n to be of t h e f o r m
9 = ~ (m, 0) ~0 (m, it).
This is s h o w n b y t h e following t h e o r e m . 1
T h e o r e m 1 . 3 3 . Let t~ (l~) be an arbitrary /unction el it which tends steadily to in/inity with L Then it is possible to lind irrational numbers
0/or
which the asser- tion 'there is a /unction(O, it) = S~ (0) ~ (it) such that, when it is chosen, the inequality
I(nO)
--c~ I < T/itis satis/ied, lor every a, by some n less than O' is /alse.
S u p p o s e t h a t t h e a s s e r t i o n in q u e s t i o n is t r u e . T a k i n g a ~ I/it, we see t h a t
( I . 331) o < (nO) < 21it
f o r some n less t h e n O.
L e t p~/q~ be t h e v-th c o n v e r g e n t to t h e simple c o n t i n u e d f r a c t i o n i + i + i +
a, a~ as
w h i c h r e p r e s e n t s 0, so t h a t Pi = I , q~ ~ a~; a n d let us c o n s i d e r t h e s y s t e m of ' i n t e r m e d i a t e c o n v e r g e n t s '
p2n,~ pe,~+rp2,+l, (o < r < a2n+2), q~n,r q2n +rq2n+l
1 In proving a result of this negative character we may evidently confine ourselves to the special case in which m = I.
176 G.H. Hardy and J. E. Littlewood.
interoalated between p~./q~, and p2.+~/q2.+2. These fractions are all less t h a n 0 a n d increase with r. Also
(1.332 ) 0 ~'*'" a'~.+z--r
q 2 n , r q2n, r q 2 n + 2
where a's.+2 is the complete quotient corresponding to a=.,+2, and q ' 2 . +2 ~ a r 2 n + 2 q2u + ! + q2n*
L e t
(I 9 333) Z . ~ 2 q'=.+2
a ~ 2 n + 2 - - 8 '
where s is a particular value of r which we shall fix in a moment. We shall suppose a2.+2 large, a n d s also large, b u t small in comparison with a2.+~. In these circumstances i~. will be approximately equal to 2 q2~+~.
We shall now prove t h a t if ( i . 334)
t h e n
(~ .335)
o < (Q O) < 2/)..
Q > qsn,9
From ( I . 3 3 4 ) it follows t h a t there is a fraction P/Q such t h a t ( z . 336)
On the other h a n d (z .337)
P 2
Z:Q"
q2 n,8 ~'. q~., 9
If P/Q actually gave a b e t t e r approximation b y defect to 0 t h a n p2.,,/q=.,8, it would follow at once t h a t Q > 92.,9 We m a y therefore suppose the c o n t r a r y ; and then it follows from (1.335) and (i .337) t h a t
Hence
o < p2,,,9 P 2
But
0 < ~ . , , Q - - q,.,, P < 2 g2.,,/L,.
and
Some problems of Diophantine Approximation.
I g~ a t 2 . + 2 q2~+l + q2~> q2~+l,
2 at2 n + 2 - - 8
q,~n,. = q~,~ + 8 q 2 n + l < (s + I ) q2n § Hence p2,,,~Q--q2n,~P is less than s + I , and so
1~,,,~Q--q~n,~P=~ (O < Q S s).
( I . 338)
On the other hand
and so
P 2 n , s q 2 n , s - - o . - - q 2 n , * ~Pen, s - - o = ~ ;
177
P2n,8 (Q - - q2n,8- q) = q~,~ (P - - p2~,.-o).
Hence either Q = q2~,8-~, or Q--q2,,,8-0 is divisible b y q~,~; and the latter hypo- thesis plainly involves t h a t Q > qe.,s.
On the other hand, if Q=q~,,,~_e, then P - ~ P ~ m . - o , and P a ~ , , + 2 - - s + e > 2
O - - 0 = - - - L T - - - _ _ - - - '
(Q o) > 2/).,,
which contradicts (i .337). H e n c e in a n y ease Q > q~_.,,.
It is now easy to complete the proof of the theorem. We have a /ortiori Q > s . Also, if a2~+2 is large, and s large, b u t small in comparison with a2.+2, )., will clearly be less than 4q~n+t. We m a y suppose for definiteness t h a t
s = [1/a2.+2"1.
We choose a value of 0 such t h a t the inequality
a 2 n +3 :> {tp (4 q 2 a + l ) ) a is satisfied for an infinity of values of n. Then
s > 2
B u t if Q, and a ]ortiori s, is less t h a n O, we must have
~(~0(4 q ~ . + l ) ) ' < ~2(0)tp0~. ) < ~2(0)tp (4 qe.+~);
and this is obviously impossible when n is sufficiently large. This completes the proof of the theorem.
A e t a mc~hematiea. 37. Imprim6 le 27 f6vrier 1914. ~8
178 G . H . Hardy and J. E. Littlewood.
I t should be observed t h a t the success of our a r g u m e n t depends entirely on our initial choice of a in such a w a y t h a t (n0) is small. I t would not be enough t h a t n 0 should be small, t h a t is to say t h a t (n #) should be nearly equal to either o or i : this can of course be secured b y choice of an n less t h a n O, q~
being indeed independent of 0.
i . 34. We t u r n now for a m o m e n t to the questions concerning el. If we have found a function 9 (X) which is continuous and monotonic, the inverse function is plainly a ~. The converse, however, is n o t true, and we cannot, from the existence of a ep of given form, draw a n y conclusion as to the order of a) for all values of )~.
This is clear from the fact that, to p u t it roughly, the existence of ~ asserts an inequality which need only hold very occasionally, a n d which therefore gives us information as to the behaviour of 9 only for occasional values of )~. Thus the existence of a q~ asserts much more t h a n t h a t of the corresponding ~. Since moreover it will appear (in the third paper of the series) t h a t in applications of the present t h e o r y it is always the properties of q), a n d not those of ~, which are relevant, we are justified in regarding theorems concerning ep as of r a t h e r minor importance. There are, however, one or two results which are worth noticing, a n d which are not deductions from t h e corresponding results concerning O. I t should be observed t h a t whereas we wish ~) to increase as slowly as possible, we wish ep to increase as rapidly as possible.
Theorem 1. 340. I t is possible to choose the a's so that ep (m, 0, a, n) increases with arbitrary rapidity. Moreover the a' s m a y be chosen in an arbitrarily small neigh-
bourhood o/ any set (al, a 2 , " " a,n).
We omit the proof of this theorem, which is e a s y .
Theorem 1. 341. I / k ~ i and m ~ i , then, provided Only that 0 is irrational, we m a y take
I ( n ) = - n
3 (a /unction independent el both 0 and a).
This follows at once from the a r g u m e n t (a) of x. i i . I t is n a t u r a l to sup- pose t h a t , when m > i, we m a y t a k e
where co (m) depends only on m. B u t this we have n o t been able to prove.
A comparison of Theorems 1 . 3 3 a n d 1 . 3 4 1 shows v e r y clearly the differ- once between theorems involving q} and those involving r a n d the greater d e p t h a n d difficulty of the former.
Some problems of Diophantine Approximation. 179 1 . 3 5 . T h e o r e m 1 . 3 3 shows t h a t it is hopeless t o e x p e c t a n y s u c h simple r e s u l t c o n c e r n i n g 9 as is a s s e r t e d c o n c e r n i n g 9 in T h e o r e m 1 . 341. I t is h o w e v e r possible t o o b t a i n t h e o r e m s w h i c h i n v o l v e q~ a n d c o r r e s p o n d t o T h e o r e m 1 . 341, if we s u p p o s e t h a t c e r t a i n classes of i r r a t i o n a l s (as well as t h e r a t i o n a l s ) a r e e x c l u d e d f r o m t h e r a n g e of v a r i a t i o n of 0. In t h e t w o t h e o r e m s which follow it is s u p p o s e d t h a t m = i a n d k = I,
T h e o r e m 1 . 3 5 0 . Let 0 be con]ined to the class of irrationals whose partial quotients are limited, a set which is everywhere dense. Then we m a y take
T h e o r e m 1 . 351. Let 0 be confined to the class o/ irrationals whose partial quotients an sati,/y, ]rom a certain value o/ n onwards, the ineq~tality
Then we may take,
a. < n ~+~ ( 6 > 0 ) .
9 = Z (log Z) 1 + ~' .q (0) where ~r ia any number greater than &
T h e i n t e r e s t of t h e last t h e o r e m lies in t h e f a c t t h a t t h e s e t in q u e s t i o n is of m e a s u r e i, ~ so t h a t we m a y t a k e O t o be of t h e f o r m Z ( l o g X ) l + ~ ( 0 ) , ~ w h e r e e is a n a r b i t r a r i l y small p o s i t i v e n u m b e r , f o r almost all v a l u e s of O.
T h e p r o o f s of t h e s e t h e o r e m s a r e simple a n d d e p e n d m e r e l y on a n a d a p t - a t i o n of KRONECKER'S a r g u m e n t r e p r o d u c e d in I . i i . S u p p o s e first t h a t t h e p a r - tial q u o t i e n t s of 0 are l i m i t e d . W e c a n choose H so t h a t , w h e n ;t is assigned, t h e r e is a l w a y s a d e n o m i n a t o r q,~ of a c o n v e r g e n t t o 0 s u c h t h a t
( I . 35o) 2 Z < q m < H ~
W e t a k e q = qm. I t follows f r o m KRONEOKER'S a r g u m e n t t h a t t h e r e is f o r a n y . a n u m b e r v s u c h t h a t
[ ( v O ) - - a [ < 2 / q , q / 2 < v < 3 q / 2 , a n d so
I 0) - . I <
for some v less t h a n a c o n s t a n t m u l t i p l e of ~t.
t By a theorem of BORZL and BER~STZlS. See BOREL, Rendiconti di Palermo, eel. 27, p. 247, and Math. An~., vet. 72, p. 578; B~aNs~m~, Math. Ann., eel. 71, p. 417.
It is not difficult to replace ~.(log ~)1+8 by ~ log 2 (log log ~)l + e, or by the correspond- ing but more complicated functions of the logarithmic scale.
1 8 0
T h e p r o o f of T h e o r e m 1 . 3 5 1 is v e r y similar.
q , , - l <_ 2 ;~ ~ q ....
a n d so
G. H. Hard), and J. E. Littlewood.
W e s u p p o s e t h a t
q , , / m 1 + ~ < "2 )~ < q ....
T h e r e is a c o n s t a n t Q such t h a t q,, > e0-~; a n d f r o m t h e s e f a c t s it follows easily t h a t q,~ < it (log it)l + ~,
for s u f f i c i e n t l y large v a l u e s of 2. T h e p r o o f m a y n o w be c o m p l e t e d in t h e s a m e m a n n e r as t h a t of T h e o r e m 1 . 350.
I t is n a t u r a l t o s u p p o s e t h a t t h e s e t h e o r e m s h a v e a n a l o g u e s w h e n m > I. B u t o u r a r g u m e n t s , d e p e n d i n g as t h e y d o on t h e t h e o r y of c o n t i n u e d f r a c t i o n s , d o n o t a p p e a r to be c a p a b l o of e x t e n s i o n .
9 4- - - T h e g e n e r a l s e q u e n c e ( [ ( n ) O ) a n d t h e p a r t i c u l a r s e q u e n c e
(a"O)
i . 4o. W e r e t u r n n o w t o t h e general s e q u e n c e (] (n) 0): it will be c o n v e n i e n t to w r i t e itn f o r [ (n). W e s u p p o s e t h e n t h a t (An) is a n a r b i t r a r y i n c r e a s i n g s e q u e n c e of n u m b e r s whose l i m i t is i n f i n i t y . 1
I t would be n a t u r a l to a t t e m p t t o p r o v e t h a t , if 0 is i r r a t i o n a l a n d a is a n y n u m b e r s u c h t h a t o _< ~ < I , a s e q u e n c e (n~) c a n be f o u n d such t h a t
( ~ 0) --- ~;
b u t we saw in I . o o t h a t this s t a t e m e n t is c e r t a i n l y false, f o r e x a m p l e w h e n
)~n---2 n o r A n ~ !
T h e r e s u l t which is in f a c t t r u e was s u g g e s t e d to us b y a t h e o r e m of B~.RN- ST~,I~, s which r u n s as follows:
I [ )~, is a l w a y s a n integer, then the set o/ values o/ 0 /or w h i c h
( Z . 0 ) - - o
is o~ m e a s u r e zero.
This result, when c o n s i d e r e d in c o n j u n c t i o n with w h a t we h a v e a l r e a d y p r o v e d , a t once suggests t h e following t h e o r e m .
9 In the introductory remarks of I. oo we stated our main problem subject to the restrict- ion that ~-n is an integer. No such restriction, however, is required in what follows.
g F. B~RI~S'rEIN, lOC. tit.
