SOME PROBLEMS OF DIOPHANTINE APPROXIMATION
BY G. H. HARDY AND J. E. LITTLEWOOD.
1. Let us denote by [x] and (x) the integral and fractional parts of the real
number x, so that
(x) = x — [x], 0 ^ (x) < 1.
Let 0 be an irrational number, and a any number between 0 and 1 (0 included).
Then it is well known that it is possible to find a sequence of positive integers
w<i, n2, n3, ... such that
(nr0) —>a
as r —» oo . Now let f(n) denote a positive increasing function of n, integral when n
is integral, such as
n, n2, n\ ..., 2n, Sn, ...,n\, 2n\ ..., 22", ...,
and let fr denote the value of f(n) for n = nr. The result just stated suggests the
following question, which seems to be of considerable interest :—For what forms of
f(n) is it true that, for any irrational value of 0, and any value of a such that
0 — a < 1, a sequence nr can be found stich that
It is easy to see that, when the increase of f(n) is sufficiently rapid, the result
suggested will not generally be true. Thus, if f(n) — 2n, and 0 is a number which,
when expressed in the binary scale, shows at least h O's following upon every 1, it is
plain that
(2»0)<£ + X*,
where \& 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 -| as we please. If f(n) = Sn we can obtain an excluded
interval whose length is as near | as we please, and so on, while if f(n) — nl it is
(as is well known) possible to choose 0 so that (n\ 0) has a unique limit. Thus
(nle)-*0.
2. The first object of this investigation has been to prove the following
theorem :—
Theorem 1 . If f (n) is a polynomial in n, with integral coefficients, then a
sequence can be found for which (/ r ö)—>a.
224
G. H. HARDY AND J. E. LITTLEWOOD
We shall give the proof in the simple case in which
f(n) = 7i2,
a case which is sufficient to exhibit clearly the fundamental ideas of our analysis.
Our argument is based on the following general principle, which results from the
work of Pringsheim and London on double sequences and series* :
•
J
V
r
,
s
ì
< P r ,
s
j
are a finite number of functions of the positive integral variables r, s ; and if
lim lim fTt s — a, lim lim <£r>8 = b, ... ;
then we can find a sequence of pairs of numbers
(rl9 Sj), (r2, So), (r3, s3) ...
such that ri —» GO , S{ —» oo and fri^i —> a, <£^,s{—>b, ..., as i —» oo .
We shall first apply this principle to prove that a sequence nr can be found
so that
(w r 0)-+O, {n2r0)-*O
simultaneously. We shall, in the argument which follows, omit the brackets in (nr0),
etc., it being understood always that integers are to be ignored.
We can choose a sequence nr so that nr0-*O.
The corresponding values n\0
are infinite in number, and so have at least one limiting point £ ; f may be positive
or zero, rational or irrational. We can (by restricting ourselves to a subsequence of
the nr9&) suppose that
wr-0->O,
n\6->%.
If £ = 0, we have what we want. If not we write
A s = (nr+ n8) 0, </vs 8 = (nr + 7is)2 0.
lim lim / r j g = lim ns0=^O,
Then
lim
lim < £ n s = l i m (£ + n2s0)=* 2%.
s -*- co r ->• oo
s -*• oo
Hence, by the general principle, we can pick out a new sequence pr such that
pr0-*0,
p\0->2%.
Repeating the argument, with nr+p8 in the place of nr + ns, we are led to a
sequence qr such that
qr0->O,
q\0->3£;
and it is plain that by proceeding in this way sufficiently often we can arrive at
a sequence nTilc such that
nr, k 0 --> 0, n\ & 0 —>• kg,
for any integral value of k.
Now whatever number £ is, rational or irrational, we can find a sequence ks
such that
fc,f->0
as s —» oo .
Then
lim
lim nrtîCs 0 = lim 0 = 0 ,
s -»• oo r -*• co
s ->• co
lim
lim n2rtJcs 0 = lim ks^= 0.
S-^-cc
V -*• 00
5-3^GO
* Pringsheim, Sitzungsberichte der k. b. Akademie der Wiss. zu München, vol. 27, p. 101, and Math.
Ännalen, vol. 53, p. 289 ; London, Math. Annalen, ibid., p. 322.
SOME PROBLEMS OF DIOPHANTINE APPROXIMATION
225
Applying the general principle once more we deduce a sequence of values of n for
which (n0) -» 0, (n20) -» 0 simultaneously.
When we have proved that there is a sequence nr for which n2r0—>O, it is very
easy to define a sequence vrnr, where vr is an integer depending on r, which gives
any arbitrary a as a limit. We thus complete the proof of Theorem 1 in the case
f(n) = n2. An analogous method may be applied in the case of the general power nk.
As in the course of this proof we obtain a sequence for which
w0->O, n20-+O,
..., w*0->O
simultaneously, we thus prove the theorem when a = 0 for the general polynomial f(n).
The extension to the case a > 0 may be effected on the same lines as in the case
f(n) = nk, but it is more elegant to complete the proof by means of the theorems of
the next section.
It may be observed that the relation
n0->O
may be satisfied uniformly for all values of 0, rational or irrational ; that is to say,
given any positive e, a number N (e) can be found such that
n0 < e
for every 0 and some n, which depends on e and 0 but is less that N (e). Similar
results may be established for n20, n*0,
The chief interest of this result lies in
the fact that it shows that there must be some function $ (n), independent of 0, which
tends to zero as n —> oo and is such that for every 0 there is an infinity of values of n
for which
n20<$(n)*.
3. The following generalisation of the theorem quoted at the beginning of § 1
was first proved by Kronecker f :—
If 0, $, A|T, ... are any number of linearly independent irrationals (i.e. if no
relation of the type
a0 + 6<jf>+ c*/r-f ... = 0 ,
where a,b,c, ... are integers, not all zero, holds between 0, <j>, -vjr, ...), and if a, /3, 7, ...
are any numbers between 0 and 1 (0 included), then a sequence nr can be found
such that
nr0 —> a, wr<£>—>/3, nrty—*<y,
This theorem, together with the results of § 2, at once suggest the truth of the
following theorem:—
Theorem 2 .
If 0, cj),^, ... are linearly independent irrationals, and
«j, ft, 7 , , . . . ( $ = 1 , 2 ,
...,*)
* It is well known that, in the case of n&, <f> {n) may be taken to be ljn. No such simple result holds
when a > 0 : exception has to be made of certain aggregates of values of 6. On the other hand, if 6 is a
fixed irrational, the relation nß -*• a holds uniformly with respect to a. All these results suggest
numerous generalisations.
f Werke, vol. 3, p. 31. The theorem has been rediscovered independently by various authors, e.g. by
Borei, F. Riesz, and Bohr (see for example Borei, Leçons sur les séries divergentes, p. 135, and F. Riesz,
Comptes Rendus, vol. 139, p. 459).
M. c.
15
226
G. H. HARDY AND J. E. LITTLEWOOD
k sets of numbers all lying between 0 and 1 (0 included), then it is possible to find
a sequence of values of n for which
n0 —>.oti, n§ —>ßly ny\r -J>y1, ...,
n20-+a2, ?i2</>->/32, n2y]r-*y2, ...,
nk0-*OLk, nk<f)-+ßk, nkyjr-^>yk, ....
This theorem we prove by means of two inductions, the first from the case of k sets
a
i> ßi> yi> ••• to the case of k +1 sets m which the numbers of the last set are all zero,
the second from this last case to the general case of k + 1 sets. The principles which
we employ do not differ from those used in the proof of the simpler propositions
discussed in § 2.
4. The investigations whose results are summarised in the preceding sections
were originally begun with the idea of obtaining further light as to the behaviour of
the series
from the point of view of convergence, summability, and so forth. If we write*
sn®= 2 d'-iVM 8n®= x evH,i} Sn(4)= x (_ i y - i e*2**
v^n
v<n
v <n
{2)
it is obvious that, if sn is any one of sn , ..., then sn — 0 (n). If 0 is rational, either
sn = 0 (1) or sn = An + 0 (1), where A is a constant : the cases may be differentiated
by means of the well known formulae for " Gauss's sums." Similar remarks apply to
the higher series in which (e.g.) v2 is replaced by v3, v4, .... The results of the
preceding sections have led us to a proof of
Theorem 3 . If 0 is irrational, then sn = o (n) : the same result is true for the
corresponding higher sums.
The argument by which we prove this theorem has a curious and unexpected
application to the theory of the Riemann ^-function ; it enables us to replace Mellin's
result f (1 + ti) = 0 (log 11 |)f by
£ ( l +fo")= o ( l o g | t | ) .
Theorem 4 . Theorem 3 is the best possible theorem of its kind, that is to say
the o (n) which occurs in it cannot be replaced by 0 (n^>), where <£ is any definite
function of n, the same for all 0's, which tends to zero as n —> oo.
But although Theorem 3 contains the most that is true for all irrational 0's, it
is possible to prove much more precise results for special classes of 0's. Here we use
methods of a less elementary (though in reality much easier) type than are required
for Theorem 3, the proof of which is intricate.
In Chap. 3 of his Galcul des Résidus^ M. Lindelöf gives a very elegant proof of
the formula
*% evH™lq= /(llY% e-v*q*ilp
* The notation is chosen so as to run parallel with Tannery and Molk's notation for the ^-functions : n
is not necessarily an integer.
t Landau, Handbuch der Lehre von der Verteilung der Primzahlen, p. 167,
$ pp. 73 et seq.
SOME PROBLEMS OF DIOPHANTINE APPROXIMATION
227
of Genocchi and Schaar. Here p and q are integers of which one is even and the
other odd. By a suitable modification of Lindelöf's argument, we establish the
formula
0(1)
V0 '
where 0 is an irrational number, which we may suppose to lie between —1 and 1,
A, is one of 2, 3, 4, \ a corresponding one of the same numbers, and 0 (1) stands for
a function of n and 0 less in numerical value than an absolute constant.
'L»=VÖV«K
We observe also that the substitution of 0 -f 1 for 0 merely permutes the indices
2, 3, 4, and that the substitution of — 0 for 0 changes sn into its conjugate. If now
we write 0 in the form of a simple continued fraction
I I I
a1 -j- a2 -f az + ... '
1
1
0 = «1 + —,
01 ' 0j
«2 + 02 '
and put
we obtain
'wVö) #<"*>+ V0
V U -£iw+o(i)f4 +
V
^L(-^) + 0(l)i4+.77^
IV0 V(00i) +v/.(00102)j
and so on. We can continue this process until n00x02... < 1, when the first term
vanishes, and we are left with an upper limit for | sn | the further study of which
depends merely on an analysis of the continued fraction.
We thus arrive at easy proofs of Theorems 3 and 4 for k = 2. We can also prove
Theorem 5 . If the partial quotients an of the continued fraction for 0 are
limited* then sn (0) — 0 (\Jn). In particular this is true if 0 is a quadratic surd,
pure or mixed.
5. The question naturally arises whether Theorem 5 is the best possible of its
kind. The answer to this question is given by
Theorem 6 . If 0 is any irrational number, it is possible to find a constant H
and an infinity of values of n such that
\sn(0)\>H^n.
The same is true of all Cesàro's means formed from the series.
The attempt to prove this theorem leads us to a problem which is very interesting
in itself, namely that of the behaviour of the modular functions
2
q
as q tends along a radius vector to an " irrational place " eQni on the unit circle. If
f(q) denotes any one of these functions, it is trivial that
t ^-iy\ s / , 2 ( - i f - y
/(3) = 0{(l-|c|)-*}-
* This hypothesis may be generalised widely.
15—2
228
G. H. HARDY AND J. E. LITTLEWOOD
If q tends to a rational place, it is known that f(q) tends to a limit or becomes
definitely infinite of order ^. By arguments depending upon the formulae of
transformation of the ^-functions, and similar in principle to, though simpler than,
those of § 4, we prove
Theorem 7.
When q tends to any irrational place on the circle of convergence,
/ ( 2 ) = o {(1-1 Si)'*}-
No better result than this is true in general.
irrationals defined in Theorem 5, then
f(q) =
If q-* e01Ti, where 0 is one of the
0{(l-\q\yi}.
Further, whatever be the value of 0, we can find a constant H and an infinity of values
of\q\, tending to unity, such that
\f(q)\>H{(l-\q\)-i\.
In so far as these results assign upper limits for \f(q)\, they could be deduced
from our previous theorems. But the remaining results are new, and Theorem 6 is
a corollary of the last of them. Another interesting corollary is
Theorem 8 .
The series
Zn~a < > - ^ 6 i r i , tn~a
en"dTl, 2 ( - I f n'a
emiTi,
luhere 0 is irrational, and a è \, can never be convergent, or summable by any of
Cesàro's means.
On the other hand, if a > ^, these series are each certainly convergent for an
everywhere dense set of values of 0. They are connected with definite integrals of
an interesting type : for example
t
i
(_!)»-:
n
eH
' \/ ( " ) I
e ix2
~
'°S (^ c o s 2 ma^ ^x'
where w = sl(0ir), whenever the series is convergent.
6. We have also considered series of the types S(?i0), %(n20), ....
convenient to write
Iw0} = ( w 0 ) - i , * w = 2 {1/0}.
It is
v 5^ n
Arithmetic arguments analogous to those used in proving Theorems 3 and 4
lead to
Theorem 9 . If 0 is any irrational number, then sn — o(n). The same result
holds for the series in which v is replaced, by v2, v3, ..., vk, ...*. Further, this result is
the best possible of its kind.
* This result, in the case fc = l, has (as was kindly pointed out to us by Prof. Landau) been given by
Sierpinski (see the Jahrbuch über die Fortschritte der Math., 1909, p. 221). Similar results hold for the
function
which reduces to {#} for a = 0.
SOME PROBLEMS OF DIOPHANTINE APPROXIMATION
229
When k — 1, we can obtain more precise results analogous to those of §§ 4, 5.
The series 2 [n0] behaves, in many ways, like the series 2e^20™. The rôle of the
formula of Genocchi and Schaar is now assumed by Gauss's formula
2
vp +
2
.7. i
odd integers. Taking
i
where p, q are
prove Theorem 9 in the case k = 1.
.^l = i(p-i)(î-i),
this formula as our starting point we easily
Further, we obtain
Theorem IO. If 0 is an irrational number of the type defined in Theorem 5,
then sn=0 (log n).
This corresponds to Theorem 5. When we come to Theorem 6 the analogy
begins to fail. We are not able to show that, for every irrational 0 (or even for
every 0 of the special class of Theorem 5), sn is sometimes effectively of the order of
log n. The class in question includes values of 0 for which this is so, but, for anything we have proved to the contrary, there may be values of 0 for which sn= 0 (1).
And when we consider, instead of sn, the corresponding Cesàro mean of order 1, this
phenomenon does actually occur. While engaged on the attempt to elucidate these
questions we have found a curious result which seems of sufficient interest to be
mentioned separately. It is that
2 {V0}2 =
j\n+O(l)
V< w
for all irrational values of 0. When we consider the great irregularity and obscurity
of the behaviour of 2 \y0\, it is not a little surprising that 2 [v0]2 (and presumably
the corresponding sums with higher even powers) should behave with such marked
regularity.
7. The exceedingly curious results given by the transformation formulae for
the series 2e™207™, 2 {n0\ suggest naturally the attempt to find similar formulae for
the higher series. It is possible, by a further modification of Lindelöf s argument,
to obtain a relation between the two sums
n
m
tev*07ri,
tfjL-ie-^
,
™,
where K = \/(32/270). The relation thus obtained gives no information about the
first series that is not trivial. We can however deduce the non-trivial result
l
Similar remarks apply to the higher series 'Ze71*6'*1 and to the series 2 {^0}, where
k>l.
But it does not seem probable that we can make much progress on these lines
with any of our main problems.
In conclusion we may say that (with the kind assistance of Dr W. W. Greg,
Librarian of Trinity College, and Mr J. T. Dufton, of Trinity College) we have
tabulated the values of (n20) for the first 500 values of n, in the cases
0=^
vio
= -31622776..., 0 = e.
The distribution of these values shows striking irregularities which encourage a
closer scrutiny.