Topological dynamics and combinatorial number theory

TOPOLOGICAL

DYNAMICS NUMBER

AND

COMBINATORIAL

THEORY By

H. F U R S T E N B E R G A N D B. WEISS

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Introduction

A number of results in combinatorial number theory have accumulated having the general form: For any finite partition of the natural numbers N = {1, 2, 3, 99-} into

classes {CI, C2," 9 ", Cr}, at least one of the classes possesses property P. For example, if P is the property that a subset contains arithmetic progressions of arbitrary finite length, then the foregoing becomes the well known theorem of van der Waerden. We shall discuss here a collection of such results, presenting a new approach, based on topological dynamics. There will be very little new by way of combinatorial results, but we hope that the machinery we develop will prove useful in tackling other problems in combinatorial number theory as well as suggesting problems of independent interest in the theory of dynamical systems. This work originated in certain investigations into the recurrence properties of dynamical systems and also in the investigations in [4] relating Szemer6di's theorem on arithmetic progressions to an ergodic-theoretic theorem. In all of this we have benefited greatly from many discussions with S. Glasner. Apparently the first result of the type we are considering is due to D. Hilbert [5] who needed it in proving his "irreducibility theorem". To formulate Hilbert's result it will be useful to define "parallelepideds". A subset of a group is a parallelepiped

of dimension d if it has the form A = A~ U A2 tO 99 9 tO Ad where A~ = {a, b}, a and b any two elements, A2 is a translate of A1, A2 = A ~ + cl, A 3 = (AIUA2)+c2,'",A~ Theorem

= (A~toA2to " - tOAa-I)+ca-1, I A f = U .

0.1 (D. Hilbert). For any finite partition N = C1 to C2 U 9 9 9 U Cr

and any d >- 1, there exists an index j and a parallelepiped A of dimension d, such that Cj contains infinitely many translates of A . The next result of this variety was obtained by I. Schur in connection with Fermat's last theorem. By means of it one can show that for any m there exist 61 JOURNAL D'ANALYSE MATHISMATIQUE, Vol. 34 (1978)

62

H. FURSTENBERG AND B. WEISS

non-trivial solutions to the congruence x " + y " = z " ( m o d p ) ciently large prime. (See [9] for details.) Theorem

when p is a suffi-

0 . 2 (I. Schur), For a n y finite partition N = C1 U C~ U " 9 O Cr one

can l l n d two numbers u, v in one o f the classes ~ such that u + v also belongs to Cj.

The subject came to Fife when following an afternoon's discussion of Baudet's conjecture with E. Artin and O. Schreier [11], B. van der Waerden proved [10]: Theorem

0.3 (van

der

Waerden). For

any

finite

partition

N =

C~ U C2 U 9 9 9 U C, there is a C~ containing arithmetic progression o f arbitrary finite length.

This was extended by I. Schur, A. Brauer and R. Rado. The following result of Schur combines both of the foregoing theorems [1]. Brauer). For a n y finite partition N = C1 U C2 U . 9 9 U C, there is a Ci such that f o r a n y t = 1,2, 3 , - - - there is a n u m b e r Theorem

0 . 4 (I.

Schur,

A.

d E C~ a n d a n u m b e r e such that the arithmetic progression e + id, 0 < i 0 there exists a finite set o[ transformations S~, $2,'" ", SN ~ G such that for any x, y ~ X , min,d(Six, y) < e.

P r o o f . If V is any open subset of X, U s ~ S - 1 V

is an open G-invariant set

TOPOLOGICAL

65

DYNAMICS

which by minimality is all of X. Since X is c o m p a c t , a finite subcovering c o v e r s X. Letting V range over a finite cover of X by sets of diameter < e we o b t a i n t h e condition of the lemma. T h e converse is clear. Proposition

1.2. Let (X, T) be a dynamical system with X a compact metric

space and let A be a homogeneous closed subset of X. Suppose that for every e > 0 we can find x, y E A and n >-_1 with d(T"x, y ) < e, then for every e > 0 z ~ A with d ( T " z , z ) < e, for some n >=1. Proof.

we can f i n d

A s s u m e that, as stated, for every e > 0 we can find s o m e n => 1 with

d(T"x, y) < e for some pair of points x, y ~ A. W e claim that the point y can in fact be chosen arbitrarily in A. F o r let G be the g r o u p acting minimally on A a n d c o m m u t i n g with T, and let S,, $2, 9 9 ", S~, E G satisfy

(1.1)

m i n d ( S , x , y ) < e/2, i

x,y EA.

N o w choose xo, yo and no so that d(T"~

y o ) < 6 where 6 is so small t h a t d(x, x') < 6 implies all d(Sjx, Six' ) < e/2. T h e n d(SjT"~ Sjyo) = d(T"~ sjyo) < e/2, and combining this with (1.1) we obtain

min d(T~'Sjxo, y ) < e J

for any y ~ A. H a v i n g established this we c h o o s e an arbitrary point Zo ~ A a n d w e find z~ E A, nl ~ N with (1.2)

d( T"'z,, zo) < e /2.

R e p e a t the p r o c e d u r e for z,, finding z 2 E A, n 2 E N with (1.3)

d( Thz2, z,) < e2

where e2 is so small that (i) e2 < e/2, (ii) if we replace zl by T"~z2, the inequality of (1.'2) is still valid. N o w p r o c e e d inductively. A s s u m e zo, z,, 9 9 zr have b e e n c h o s e n in A as well as n l , . . . , nr E N and e2,. " ' , e r with ej < e/2 and (1.4)

d(T"%, zj_,) < ei,

j = 1 , . . . , r.

W e find er+, < e/2 so that (1.4) is valid when z, is replaced by a point w h o s e distance from it is less than er+l. Then find z,+t ~ A and m+l E N with

66

H. FURSTENBERG AND B. WEISS

(1.5)

d(T"'+'zr+,,z,) x i = 1, 9 9 p, a n d as

soon as T,"kx E U for all i we o b t a i n (1.7). L e t us see h o w van d e r W a e r d e n ' s t h e o r e m follows f r o m T h e o r e m 1.4. Proof

of Theorem

0.3.

L e t N = C~ U C2 t3 9 9 9 U Cr b e a givan p a r t i t i o n of N,

let A = {1,2,- .., r} a n d f o r m I l = A z, t h e s p a c e of A - v a l u e d s e q u e n c e s . D e f i n e t h e shift S : fl---~ [ l by S w ( n ) = w(n + 1). W e e n d o w f l with a m e t r i c

(1.8)

d(w,o~')=inf{k-~:

~o(n)=to'(n )

for

Inl O,

sO(n) = 1

if n ~ O.

68

H. FURSTENBERG AND B. WEISS

Let X be the set of limit points of the sequence {S~'}.=~,2.3,... Clearly we will have S X = X . N o w set T, = S' for i = 1, 9 9 p and apply T h e o r e m 1.4 to (X, T1,-" ", Te).

There will exist some r/ E X with 1

"

d(TTrt, r t ) < ~ ,

1

d(T~'rt, r t ) < ~ ,

".',

But this m e a n s that r / ( 0 ) = r / ( n ) = r/(2n) . . . . .

d(T~r/,r/) ~ } as all the n, --9 ~. By T h e o r e m 1.4 there will exist a point r/ and a n u m b e r n E N with d ( S ~ , S2,2.. 9 . S ~~ , r/, r/) < e

for all vectors (a~, 9 9 -, a,,) E N " with l a, I ~ k, i = 1,. 9 m. F o r appropriate e this gives

TOPOLOGICAL DYNAMICS

69

"rl(O,''" ,0) = rl(aln, a 2 n , " ' , a,,n). Finally since some S b S~b 2 - " S ~ -b~ : is close to ~q we obtain

~ ( b , , b 2 , " ", b,,) = ~(bl + aln, b2+ a 2 n , ' . . , b,, + a,.n). This proves the t h e o r e m .

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Proximality, idempotents, and Hindman's theorem

In this section we take X to be a c o m p a c t metric space and T a h o m e o m o r p h i s m of X. A f u n d a m e n t a l notion in topological dynamics is that of proximality.

Definition 2 . 1 . A pair of points xl, xzG X is said to be proximal for the dynamical system (X, T) if for some s e q u e n c e { n k } C N

d(T"~x,, T"kxz)--~O. (It would be m o r e accurate to use the term positively proximal since we insist that nk > 0. For simplicity of terminology we shall not m a k e the distinction.) Let us illustrate this notion by taking X = A z, the space of all sequences with a certain finite alphabet A, and T the shift transformation in X. W e adopt the metric given in (1.8), so that two sequences are close if they agree on a large interval c e n t e r e d about 0. It is then clear that two points wl, to2 G X are proximal if and only if there are arbitrarily long intervals (nk, n~ + l~) C N with , o , ( n ) = o~2(n);

n ~ u (nk, n~ + Ik).

Next we formally define the kind of set of integers occurring in H i n d m a n ' s theorem.

Definition 2 . 2 . A subset H C N is called an IP-sequence if there exists a sequence pl, p 2 , p 3 , " ' , p . , " " of elements in H such that H consists of all finite sums {p~,+ p~2+ " ' - + p~., i, < i2 < " ' " < i,, n = 1 , 2 , 3 , - - .}.

70

H. FURSTENBERG AND B. WEISS A n I P - s e q u e n c e is an approximation t o a semigroup. T h e initials IP refer to

" i d e m p o t e n c e " which we will see is a closely related notion. T h e y also stand for "infinite-dimensional parallelepiped" which, as r e m a r k e d in the introduction, gives a n o t h e r description of IP-sets. W e p r o c e e d with an analysis of the dynamical system (X, T). T h e set of all mappings, c o n t i n u o u s or not, f r o m X to itself is d e n o t e d X x and forms a c o m p a c t space in the p r o d u c t topology. It also f o r m s a semigroup for which the one-sided multiplication f ~ fro is continuous for any ]co, and the o t h e r multiplication f ~ fof is continuous if fo is continuous. N o w consider the points of accumulation E C X x of the subset {T", n = 1, 2 , . . . } C X x. W e have T " E C E , E T " C E for all n E Z. Fix f @ E and consider the points of accumulation of {T"f, n -- 1, 2 , . . . } . These are contained in E and on the other h a n d contain El. W e conclude that E E C E. Thus E is a s e m i g r o u p and it is called the enveloping semigroup of (X, T). E is a c o m p a c t semigroup for which multiplication on o n e side is continuous. Lemma

2.1.

I f E is a compact semigroup for which o n e - s i d e d multiplication

x ~ xxo is continuous, then E contains an idempotent, i.e., an element u with u 2 = u.

P r o o f . (See [3].) Let A be a minimal subset of E satisfying (i) A A C A , (ii) A is compact. T h e family of these is n o n - e m p t y since E itself has these properties and Z o r n ' s lemma guarantees that a minimal set of this kind exists. T a k e u ~ A. T h e n A u is c o m p a c t and also A u A u

C A u . So, by minimality, A u = A . In

particular for s o m e v E A we will have vu = u. Set A ' = {v @ A [vu = u}. By one sided continuity, A ' is closed, and clearly A ' A ' C A '. So A ' = A w h e n c e u 2 = u. L e m m a 2.1 tells us that the enveloping s e m i g r o u p of a dynamical system always contains idempotents. Lemma

2.2.

I f u E E is an idempotent in the enveloping semigroup of a

d y n a m i c a l system (X, T), then for every point x @ X, x and ux will be proximal.

P r o o f . By the definition of the t o p o l o g y on X x, if u E E and e > 0 , and xl, x2, 9 9 -, x,, are any points of X, there is a p o w e r T" such that d(T"x,, ux~) < e for i = 1 , 2 , . . . , m. In particular we can find n with d ( T " x , u x ) < e, d ( T " u x , u 2 x ) < e. But u 2= u, u2x = ux and so d ( T " x , T " u x ) < 2 e .

P r o p o s i t i o n 2 . 3 . Let (X, T ) be a d y n a m i c a l system, x E X, and let Z be the set o f limit points o f the forward orbit { T " x },eN. I f Y is any m i n i m a l set in Z , there is a point y E Y such that x and y are proximal.

P r o o f . It is seen from the definitions of E and Z that E x = Z. Let F = {s E E I sx @ Y}, then Fx = Y, F will be closed and since E F C F, F 2C F. By

TOPOLOGICAL DYNAMICS

71

L e m m a 2.1, F contains an i d e m p o t e n t u. So ux E Y and the proposition n o w follows from L e m m a 2.2. In particular, Proposition 2.3 implies that every point in a dynamical system is proximal to a point belonging to a minimal set. W e now look into the implications of this for the case of symbolic dynamical systems. W e consider the setup in the p r o o f of van der W a e r d e n ' s t h e o r e m . Let A be a finite set, say, A = {1, 2,. 9 r} and f o r m fl = A z. f~ is e n d o w e d with a metric (1.8) for which it is compact. T h e shift S: f~---* fI is defined by Sto(n) = to(n + 1). W e wish to characterize the points of f~ which belong to minimal sets for the system

(a, s). W e will use the term block to d e n o t e a finite o r d e r e d s e q u e n c e of elements of A: b = b(1)b(2)-., b(l)~A' whose length is I. A block b is said to occur in a block b' at the place t + 1 if

b(i)=b'(t+i),

l