Problems and results on diophantine

approximations
by

P. Erdös

The older literature on this subject (until about 1935) is

treated in the excellent book of Koksma [1] . The more recent
literature is discussed in a very interesting paper of Cigler and
Helmberg [2] . Unlike the above authors I by no means aim to
cover the literature completely and will mostly discuss only
problems on which I myself worked thus a more exact title would
have been "Problems and results on diophantine approximation
which have interested me" . There will be some overlap with my
paper "On unsolved problems" [3] . First I discuss some questions
on inequalities of distribution and on uniform distribution .
I. Let x1 , x 2 , • • • be an infinite sequence of real numbers in the
interval (0, 1) . Denote by Nn(a, b) the number of x i satisfying
a<xi <b, 1 <i<n .
We say that x1 , is uniformly distributed if for every
0 < a < b < 1
N,,,(a, b)
(1) lim = b-a .
n=oo n
The classical result of Weyl (see [1]) states that the necessary
and sufficient condition that the sequence x 1 , . . . should be
uniformly distributed is that for every integer k, 1 < k < oc
1 n .0 = f e - lim
(2) 2nikx
n=oo n j=1
Here I would like to ask a question which I have not yet answered
though it is perhaps very simple . Put
n
lim e 2nikxf
Ak =
j=1

* Nijenrode lecture

52
[2] Problems and results on diophantine approximations 53
A k can be infinite, but if x; = ja (mod 1) then A k is finite for
every k. Is it true that lim sup A,, = co? I expect that the answer
is yes . It is easy to see that if B k is the least upper bound of
I~ 1 e 2" ik xiI then lim k=. B k = 00 -
The discrepancy of x1, . . ., xn we define as follows : (This notion
as far as I know is due to van der Corput )
(3) D(x1 , . . ., xn ) = sup I N n (a, b)-(b-a)n~ .
OSa<b<1
Equidistribution is equivalent to D(xl , . . ., xn ) = o(n) . Van der
Corput conjectured and Mrs . Ardenne-Ehrenfest proved the
beautiful result that for every infinite sequence x 1 ,. x,
lim sup D(x1 , . . ., xn ) = oo .
n=oo
In fact Mrs . Ardenne-Ehrenfest showed that for infinitely many n
D (x l , . . . x n ) > c log log n/log log log n.

Roth sharpened this result by showing that for infinitely many n

D(x 1, . . ., xn ) > c(log n) 1 .
One can express the theorem of Roth also in the following finite
form : There is an absolute constant c so that to every sequence
x 1 , . . ., xn there is an m and an a < 1 so that
IN (0, a)-amp > c(log n)+.
Perhaps in Roth's Theorem c(log n)I can be replaced by c log n,
this if true is known to be best possible [4] .
I would like to ask a few related questions .
Does there exist an infinite sequence x 1 < x2 < . . . so that for
every 0 < a < b < 1
(4) lim sup N n (a, b) < oo?
n=oo
Denote by f(a, b) the upper limit and by F(a, b) the upper bound
of N,,(a, b) . The fact that D(x 1 , . . ., xn ) is unbounded only implies
that F(a, b) cannot be a bounded function of a and b) . On the
other hand it is not clear to me why 1(a, b) could not be a bounded
function of a and b, though this seems very unlikely .
Let Iz,J = 1, 1 < v < co be an infinite sequence of complex
numbers on the unit circle . Is it true that
n
lim sup max f lz-zi p = co?
n=oo z[=1 i=1
54 P . Erdös [3]

I would guess that the answer is yes . If this is the case it would
be of interest to estimate how fast maxlsmsn An, (A n = maxzH=1
ITZ-1 Iz-z i p) must tend to infinity .
Let w, . . . . wn be any n points on the surface of the unit sphere .
Let C be any spherical cap and denote by

(4) D(w 1 , . . ., zcn ) = max(N(C)-nx c ), D n = min D(w1 , . . ., wn )

C n l , . ., w„

where N(C) denotes the number of w's which are in C and x C

is the ratio of the surface of C with the surface of the sphere,
the maximum is to be taken with respect to all spherical caps .
One would expect that D n is an unbounded function of n, in other
words : n points cannot be distributed too uniformly on the surface
of the sphere (the situation is of course quite different on the
circle) . Perhaps this can be proved by the method of Roth, who
settles in his paper the analogous question for the square [4] .
Let z1 , z2 . . . be an infinite sequence of points in the plane .
Denote by N(zo , r) the number of z's in the interior of the circle
of center zo and radius r . Put

1(r) = max(N(zo , r)-7rr2)

where the maximum is to be taken over all circles of radius r .
Probably 1(r) is unbounded for every choice of the z's and one
would like to estimate how fast f(r) or F(r) = max o<RS ,r f(R)
tends to infinity . The method of Roth will perhaps help here
too [4] .
Let /(n) be an arbitrary number theoretic function which only
assumes the values ±1 . Is it true that to every c 1 there exists
a d and an m so that
m
g(m, d) = k=1
I f(k, d) > c 1 ?
It is perhaps even true that

max g(m, d) > c 2 • log n .

d, m
din C n .
The well known Theorem of van der Waerden [5] asserts that
for every k there exists an arithmetic progression a, a+d, . . .,
a+(Ic 1)d for which f(a) _ . . . = f(a+(k-1)d) .
Let finally 1 < a 1 < . . . < an be n arbitrary integers . Denote
n
M(a1, . . .,a,,)= max f I1-z° il, /(n) =min M(a1 , . . ., an )
z[=1 i=1
[ 4] Problems and results on diophantine approximations 55

where the minimum is taken over all sequences a 1 , . . ., a n . Szekeres

and I [6] proved
lim f (n)l/n = 1, /(n) > N/2n .
n=&.
Recently Atkinson [7] proved /(n) < exp(nI log n) (exp z = ez) .
The lower bound has not yet been improved, though we are sure
that this is possible, undoubtedly /(n) > nk for every k and
n > n o (k) . Atkinson's result is perhaps not far from being best
possible.
Weyl's criterion [2] does not give an estimation of the discrep-
ancy of a sequence . Turan and I [8] proved, sharpening a previous
result of van der Corput and Koksma [1] the following result :
Assume that for every k satisfying 1 < le < m we have
n
I ski
= I Y e2aikxji < y ( k) •
7=1

Then for a certain absolute constant C

(5) D(x 1 , . . ., x n ) < C + hI ) ~.

(M+1

Koksma and Szüsz independently extended this result for the

r-dimensional case [9] .
An interesting special case of our Theorem is obtained if we
assume
(6) Is k I < kA for all k < nuiA .
From (5) we obtain that (6) implies
(7) D(x 1 , . . ., x n ) < cj nAIA+ 1

We could not decide whether the error term in (7) is best possible .
Another result on the discrepancy of points in the complex
plane due to Turan and myself states as follows [10] : Let /(z) _
ao + . . . +an zn be a polynomial, denote its roots by

z„=r,eiv, ,1 <r<n,M=maxif(z)i .
IZI=1
Then for 0 < a < ~ < 2,7 we have

(n log M )
(8) 1- n < 16 .
asp sfl 2r ao a n

It would be interesting to investigate whether (8) remains true

if n denotes the number of non vanishing terms of the polynomial
56 P. Erdös [5]

ao+ . . .+ a,zkn (or if (8) does not remain true now does it have
to be modified) .
The following questions have as far as I know not yet been
investigated . Let w1 wn be n points on the unit sphere
chosen in such a way that (wi-w, denotes the distance of wi
and w,)
I w-
1<i<7<n

is maximal . Is it then true that D n = o(n)? (see (4)) . Can one

improve this estimate?
An analogous question would be the following : Put
n
(9) A n = min max IT w-wi
w l, . ., w~ w i=1

where the maximum is taken over all points w of the unit sphere
and w 1 , . . . wn varies over all n-tuplets of points on the unit
sphere. Two questions can be asked . First of all let w 1 , . . ., wn
be one of the sets for which there is equality in (9) . Is it true that
for this set Dn = o(n)? Secondly assume that maxw 11 i=1 w-w i
is not much larger than A n, how can one estimate D n?
II. Now we discuss some questions on uniform distribution .
It follows early from (2) that for every k and every irrational
number a (nka) = nka-[n ka] is uniformly distributed, this
beautiful and important result was first proved by Weyl and
Hardy-Littlewood [1] . For general sequences n1 < n 2 < . . . it is
very difficult to decide whether (n ix) is uniformly distributed e .g.
Vinogradoff [11] only recently proved that (p n m) is uniformly
distributed for every irrational a(p 1 = 2 < P2 < . . . is the
sequence of consecutive primes) . Weyl proved that for every
sequence of integers n1 < n 2 < and for almost all a (n i a) is
uniformly distributed . Sharpening previous results Cassels and
independently and simultaneously Koksma and I [12] proved
that for almost all a the discrepancy of Xk = ( nk (X) satisfies for
every a > 0
(10) D(x l , . . ., xN ) = o(Nt(log N)5/2+E) .
Koksma and I use (5), Cassels's method is more elementary .
It would be very interesting to investigate to what extent (10)
can be improved . Possible o(NI (log N)512+') can be replaced by
a(M (log log N)c) for a certain constant c . In the special case
where the sequence ni is lacunary i .e . where it satisfies ni+11ni >
c > 1 . Gal and I proved this, but our proof which is similar to the
[ 6] Problems and results on diophantine approximations 57

one we used to establish the law of the iterated logarithm [13]

for lacunary sequences n z , was not published . It is well known and
has been perhaps first obtained by Kac and Steinhaus that for
lacunary sequences (n z a) behaves as if they would be independent .
Thus our result with Gal gives no indication what will happen if
the condition nz+1/ni > c > 1 is dropped .
The following beautiful conjecture is due to Khintchine [14]
Let E be measurable subset of (0, 1) of measure m(E) . Denote

fn(a) _ I 1
1<k<n

where the summation is extended over those k's for which (koc)
is in E . Then for almost all a and every E
(11) lim fn (a)/n = m(E) .
n=oo

Presumably the same result holds if u 1 < u2 < . . . is any sequence

of integers and fn (a) denotes the number of indices k for which
(n,a) is in E . This conjecture of Khintchine is very deep, directly
or indirectly it inspired several papers . More generally one could
ask the following question : Let n1 < . . . be an infinite sequence
of integers and I (x) is any Lebesgue integrable function in (0, 1) .
Under what conditions on I (x) and on the sequence n 1 < . . . is
it true that for almost all a ((nk a) = n k a- [nk a] )

( 12 ) lim N kI/( (nka)) = f 0

f(x)dx .

Raikov proved that if n k = ak, (a > 1 integer) then (12) holds

for every integrable I (x) . A simple proof of this result using
ergodic theory was given by F . Riesz [15] .
Let nk+1 > (1+c)nk , (c > 0) and let f(x) be in L 2 and let ~ n (f)
be the n-th partial sum of the Fourier series of I (x) . Sharpening a
previous result of Kae, Salem and Zygmund I proved that if

( 13 ) fo (f(x)-~n(/(x) )) 2dx = 0
(loglog
b b n)'+')

then (12) holds [16] .

Further I constructed [16] a lacunary sequence n 1 < n2 < . . .
and a function f(x) which is in L,, for every p and for which (12)
does not hold. In fact for our I (x) we have for almost all a
N
(14) lint sup - :E f((nk a)) = oo,
N=00 N k=1
58 P . Erdös [7 ]

in fact for our I (x) we have for every e > 0 and almost all a
1 N
(15) lim sup 1-6 1 f((n k a)) = 00 .
N=c N (loglog N)~ - k=1

On the other hand I can show that if I (x) is in L 2 and {n k } is any

lacunary sequence then for almost all a and every r > 0
1 N
(16) lim , = f (nka) 0.
N-~ N(log N)1+E

There is a considerable gap between (15) and (16). I think (15)

is closer to the truth but I cannot prove this . I would also think
that (13) remains true if o(1/(log log n) 2 +E ) is replaced by
0(1/(log log log n)° ) for a certain c > 0, but I have not been able
to decide this. It is possible that (12) holds for all bounded func-
tions and every lacunary sequence {nk } . It seems impossible to
modify my example so that it should become a bounded function .
My lacunary sequence for which (12) does not hold is very
special, it would be interesting to try to determine for what
lacunary sequences (12) hold for all f(x) in L 2 (or in L 1 ) and for
which lacunary sequences this is not the case, e .g. let a > 1 be
any real number does (12) hold for the sequence [ak]? (If a is an
integer this is the quoted result of Raikov) .
Koksma [17] proved the following result : Let f (x) be in L 2 and
let {ck } be the sequence of its Fourier coefficients . Assume that
°° 1
1 ck -- < co.
k=1 djk d

Then for almost all a

(17) lim 1
n=c
I f((ka))
n k=1 =J 01
f(x)dx .

I was unable to find an I (x) in L 2 or even in L 1 for which (17) does

not hold .
III . A sequence x 1 , x2, . . . in the interval (0, 1) is said to be
well distributed if to every r > 0 there exists a ko = ko (s) so that
for every k>ko ,n>OandO<a<b<1
IN-,-+k (a, b)/n-(b-a)I < s
where Nn,n+k ( a, b) denotes the number of x,n ' s, n < m < n + k in
the interval (a, b) . As far as I know the notion of well distrib-
uted sequences was introduced by Hlawka and Petersen [18] . Let
[s] Problems and results on diophantine approximations 59

n,,,/n, > R > 1, in contrast to the result of Weyl I proved that

for almost all a the sequence (n k a) is not well distributed .' If
nk+1/nk -* oo it is not difficult to show that the values of a for
which (nk a) is well distributed has the power of the continuum .
Further I can prove that there is an irrational number a for which
(p n a) is not well distributed (compare [11]) . The proof of these
results is not yet published . It seems very probable that (p,n a) is
never well distributed (i .e . for no value of a) but I have not been
able to show this .
IV . Finally I would like to discuss some results on diophantine
approximations . Khintchine [19] proved that if /(q) is monotone
decreasing then the condition

(18) /(q) - 00
q=1 q
is necessary and sufficient that for almost all a the inequality

a- p- < /(q)
q q2
should have infinitely many solutions in integers p and q . It is
easy to see that if (18) does not hold (i .e . if jQ 1 l(q)/q < oo)
then without any assumption of monotonicity on /(q) it follows
that for almost all a (19) has only a finite number of solutions .
The question now remains : Does (18) imply (19) without any
further assumptions on /(q)? Duffin and Schaeffer and Cassels
deduced (19) from (18) under much weaker assumptions then
monotonicity of /(q), but they both showed (18) does not imply
(19) without some condition on /(q) [20] .
In his paper [20] Cassels introduced a property of sequences which
seems to me to be of interest in itself . Let n, < n 2 < . . . be an
infinite sequence of integers . Denote by 92(n,, . . ., nk_ 1 ; nk )the
number of integers 1 < a < n k for which a/n, :A b/n ; for every
1 < j < k. Clearly

9,(n l , . . ., nk_1 ; nk) > 9, (n,) .

Cassels calls the sequence {nk } a I sequence if
(1 k n1 , . . ., n z l ; n2) > 0
(20) lim inf .
k k t=1 nz
Cassels shows that there are sequences which are not I-sequences

1 Petersen informs me that this was known to him .

60 P . Erdös [91

(i .e . for which the lim inf. i n (20) is 0) . I have not succeeded to

decide the question whether there is a sequence n1 < n2 <
for which
/
1 k n t , . . ., ni-1 ; n i) \
(21) lim ' I = 0.

k=~ \ ~° i=1 ni /
I would guess that such a sequence does not exist . I can only prove
that
g,(n t , . . ., n i -1 ; ni)
lim
i=cx ni
cannot be 0 . In fact I will outline the proof of a somewhat stronger
result : assume that
(22) lim inf T(n 1 , . . ., n i- 1 ; n i )/ni = 0
i= 3
then
(23) lim sup cp(n1 , . . ., ni_l ; ni)/ni = 1 .
i=o

Assume that (22) holds . It immediately follows from

,p(n t , . . ., n i- 1 ; n i ) > T(n i ) that there must be arbitrarily large
primes p ; so that n i - 0 (mod p ;) for suitable values of i . Assume
now that nk is the smallest ni for which n i = 0 (mod p,) . Then
if 1 < a <n k , a / 0 (mod p ;) clearly implies a/n k - 7'- b/n, for

1 < j < k, or q,(n1 , . . ., n,_ t ; nk ) > (1-1/p;)nk, which implies (23) .

Cassels shows [20] that the necessary and sufficient condition
that n 1 < n 2 < . . . should have the property that the divergence
of 1k i f (-n k )/n k implies that for almost all a

Mk < f(nk)
a-
nk nk 2
has infinitely many solutions, is that n 1 < n 2 < . . . should be
a ~7, -sequence . Cassels also shows that every sequence nk+1 >
(1 +c )n k (c > 0) is a '>7, -sequence . It seems likely that a weaker
condition will imply that a sequence is a ~7 -sequence, but as far
as I know no such condition is known .
Duffin and Schaeffer [20] made the following beautiful con-
jecture : Let eq , 1 < q < co be an arbitrary sequence of non-
negative numbers . The necessary and sufficient condition that
for almost all a the inequality

p eQ
a- - < -
q q
[10] Problems and results on diophantine approximations 61

should have infinitely many solutions in integers (p, q) = 1 is that

eg 92(q)
Q=1 q

diverges. (T(q) is Euler's (p function) . It is easy to prove the

necessity, the real difficulty is to prove the sufficiency .
I proved the following special case of this conjecture . Let s > 0
be fixed and let eq = 0 or e g = e . The necessary and sufficient
condition that for almost all a .
p
a- - < qE2 , ( p, q) = 1
q

has infinitely many solutions is that ~q 1 e g 99(q)/q2 diverges .

The proof is very complicated and has not yet been published.
My proof in fact gives the following slightly sharper result : Let
eg > 0 be a bounded sequence . Then the necessary and sufficient
condition that for almost all a,

a- p
- <E2,(p,q)= 1
q q
has infinitely many solutions is that Ya 1 e g g9(q)/q 2 diverges.
Due to the great technical difficulties of the proof I am not at
present certain whether my method gives the general conjecture
of Duffin-Schaeffer.
My result immediately implies the following theorem : Let
n 1 < n 2 < . . . be an arbitrary infinite sequence of integers. The
necessary and sufficient condition that for almost all a infinitely
many of the n i should be denominators of the convergents of the
regular continued fraction of a is that 100, go(ni)/n2 diverges .
(i .e. it is well known that if la-m i/n i l < 1/2n2, (m i , n i ) = 1 then
m i/ n i are convergent of a) . Hartman and Szüsz proved a special
case of the above result [21] . Finally I would like to state four
unrelated problems on diophantine approximation .
1 . Heeke and Ostrowski [22] proved the following theorem :
Let a be an irrational number and denote by Nn (u, v) the number
of integers 1 < m < n for which
0Cu<(ma)<vc1 .

Then if both u and v are of the form (ka) then

(24) N,,(u, v) = n(v-u)+O(1)

62 P. Erdös [1 1]

Szusz and I conjectured the converse of this theorem, i .e. if (24)

holds then u = (k 1 x), v = ( k2001 unfortunately we had not been
able to make any progress with this conjecture .
2 . Denote by S(N, A, c) the measure of those a in (0, 1) for
which
x A
a- - < 2, (x, y)=1
Y y
is solvable for some y satisfying N < y < cN .
Szüsz, Turán and I conjectured that [23]

lim S(N, A, c) = I (A, c)

N=oo

exists. What is its explicit form?

In our paper [23] we only solved a very special case of this
problem. Recently Kesten [24] strengthened our results, but the
general problem is still unsolved .
3 . Consider 0 < a < 1

1 n
f(a,n)= ~ ~( 1Ca) 2) .
log e

Is it true that f (a, n) has an asymptotic distribution function?

In other words is it true that there is a non-decreasing function
g(c), g(- co) = 0, g(+oo) = 1, so that if m[/(a, n), c] denotes
the measure of the set in a for which /(a, n) < c then
lim m[/(«, n), c] = g(c). Probably g(c) will be a strictly in-
creasing continuous function. Important recent contributions to
this problem have recently been made by Kesten, [25] but as far
as I know it is not yet completely solved .
4 . The following interesting problem is due to LeVeque : Let
a 1 < a 2 < . . . be an infinite sequence tending to infinity satisfying

ai+1/ai -* 1 . Let ai < xn < ai+1 . Put

x. - ai
yn= ,0 Cy n < 1 .
ai+1 -ai
We say that the sequence x, 1 < n < oo is uniformly distributed
mod a 1 , a 2 , . . . if y n, 1 < n < oo is uniformly distributed . Is it
true that for almost all a the sequence na, 1 < n < c is uniformly
distributed mod a 1 . . . . ? LeVeque proved this in some special
cases [26] .
[12] Problems and results on diophantine approximations 63

Added in proof : Since this paper was written the following

papers were published on the problem of LeVeque :
H . Davenport, P . Erdos and W . J. LeVeque, On Weyl's criterion
for uniform distribution, Michigan Math . Journal 10 (1963),
311-314 ;
H . Davenport and W . J . LeVeque, Uniform distribution rela-
tive to a fixed sequence, ibid 10 (1963), 315-319 and
P . Erdos and H . Davenport, Publ . Math . Inst . Hung . Acad .
(1963) .

REFERENCES

J . F. KOKSMA
[1] Diophantische Approximationen, Ergebnisse Math. u . Grenzgeb . Vol . 4
Heft 4, Berlin 1936 .

.l . CIGLER and G . HELMBERG

[2] Neuere Entwicklungen der Theorie der Gleichverteilung, Jahresbericht
D.M .V. 64 (1961), 1-50.

P . ERDÖS
[3] Some unsolved problems, Publ. Math . Inst . Hung . Acad . Sci . 6 (1961),
221-254 .

T. VAN AARDENNE-EHRENFEST
[4] On the impossibility of a just distribution, Indigationes math . 11 (1949),
264-269 .

K . F . ROTH
On irregularities of distribution, Mathematika, 1 (1954), 73-79 .

B . L . VAN DER WAERDEN

[5] Beweis einer Baudetschen Vermutung, Archiv Viskunde (2) 15 (1928),
212-216, for further problems and results in this direction see

R . RADO
Studien zur Kombinatorik, Math . Zeitschrift 36 (1933) 424-480 .

P . ERDÖS and G . SZEKERES

[6] On the product HIk_ r (1-zak), Acad . Serbe des Sci . 13 (1959), 29-34 .

F . V . ATKINSON
[7] On a problem of Erdos and Szekeres . Canadian Math . Bull . 4 (1961), 7-12 .

P . ERDÖS and P . TURÁN

[8] On a problem in the theory of uniform distribution I, II . Indigationes math .
10 (1948) 370-378, 406-413 .

J . F. KOKSMA
[9] Some theorems on diophantine inequalities, Math . Centrum Amsterdam
Scriptum 5 (1950),

P . Szüsz
On a problem in the theory of uniform distribution (written in Hungarian)
C .r . first Hungarian Math. Congress 461-472 (1952) .
64 P . Erdös [ 1 3]

P . ERDÖS and P . TURÁN

[10] On the distribution of roots of polynomials, Annals of Math . 51 (1950),
105-119 .

1 . M . VINOGRADOV
[11] Über die Abschätzung trigonometrischer Summen mit Primzahlen, Isvestija
Akad . Nauk . SSSR Ser . mat 12 (1948), 225-248 .

J . W . S . CASSELS
[12] Some metrical theorems in diophantine approximation, Proc . Cambridge
Phil . Soc . 46 (1950), 209-218 .

P . ERDÖS and J. F . KOKSMA

[13] On the uniform distribution mod 1 of sequences {f(n, 0)}, Indig . Math . 11
(1949), 299-302 .

P . ERDÖS and I. S . GAL

[13] On the law of the iterated logarithm, Indig. Math. 17 (1955) 65-84.

A . J . KHINTCHINE
[14] Ein Satz fiber Kettenbruche mit arithmetischen Anwendungen, Math .
Zeitschrift 18 (1923), 289-306, seep . 303-304 .
F . RIESZ
[15] Stir Ia théorie ergodique, Comment . Math. Helv . 17 (1945), 221-239 .

1' . ERDÖS
[16] On the strong law of large numbers, Trans . Amer. Math. Soc . 67 (1949), 51-56.

J . F . KOKSMA
[17] Estimations de fonctions a I'aide d'intégrales de Lebesgue, Bull . Soc. Math .
Bell . 6 (1953-54) 4-13.
E . HLAWKA
[18] Zur formalen Theorie der Gleichverteilung in kompakten Gruppen, Rend .
Cise . Mat . Palermo 4 (1955), 33-47 .

G . M. PETERSEN
Almost convergence and uniformly distributed sequences, quarterly J . of
Math. Oxford II Ser. 7 (1956), 188-191 .
See also F . R . Keogh, B . Lawton and G . M . Petersen, Well distributed
sequences, Canad . J . Math . 10 (1958), 572-576.

A . J . KHINTCHINE
[19] Einige Sätze fiber Kettenbrüche mit Anwendungen auf die Theorie der
Diophantischen Approximationen Math . Ann . 92 (1924), 115-125.

11 . J . DUFFIN and A . C . SCHAEFFER

[20] Khintchine's problem in metric diophantine approximation, Duke Math .
.J . 8 (1941), 243-255, .T. W . S . Cassels, Some metrical theorems in diophantine
approximation I . Proc . Cambridge Phil . Soc. 46 (1950), 209-218 .

S . HARTMAN and P . SZÜSZ

[21] On congruence classes of denominators of convergents, Acta Arith . 6 (1960),
179-184 . See also

P . SZÜSZ
Verallgemeinerung and Anwendungen eines Kusminschen Satzes, ibid . 7
(1962), 149-160, and P . SÜSZ, Über die metrische Theorie der Diophantischen
Approximationen (will soon appear in Acta Arith . ) .
 Problems and results on diophantine approximations 65

E. HECKE
[22] Über analytische Funktionen and die Verteilung der Zahlen mod 1, Abh .
math. Semin . Hamburg Univ. 1 (1922) 54-76,

A. OSTROWSKI
Miszellen IX and XVI, Notiz zur Theorie der Diophantischen Approxima-
tionen and zur Theorie der linearen Diophantischen Approximationen .
Jahresbericht der Deutschen Math . Ver . 36 (1927), 178-180 and 39 (1930),
34-46) .
P . ERDÖS, P . Szusz and P . TURÁN
[23] Remarks on the theory of diophantine approximation, Coll . Math . 6 (1958),
119-126 .

H. KESTEN
[24] Some probabilistic theorems on diophantine approximations, Trans . Amer .
Math . Soc . 103 (1962), 189-217 .

H. KESTEN
[25] Uniform distribution mod 1, Annals of Math . 71 (1960), 445-471 .

W . J . LEVEQUE
[26] On uniform distribution modulo a subdivision, Pacific J . of Math. 3 (1953),
757-771 .

(Oblatum 29-5-63) University College, London.