Ergodic sequences of measures and a problem in additive number theory

Vol. 31, 1978 21

Ergodic sequences of measures and a problem in additive number theory

By

H A R A L D N I E D E R R E I T E R

1. Introduction. Let Z be the additive group of integers in the discrete topology and let E ~ (an), n = 1, 2 . . . . . be a sequence of elements of Z. For N ~ 1, the set consisting of the first N terms of (an) is denoted by EN and the measure/~N on the Bohr compactification ~ of 7/is defined by l ~ (B) = I B n EN]/IEN [ for all subsets B of 2, where I SI denotes the cardinality of a set S. The sequence of measures /~1, ~2 . . . . is called ergodic if it converges weakly to the Haar measure on ~ (cf. [1]). Ergodic sequences of measures can also be characterized in terms of distribution properties of the original sequence (an). In detail, this characterization says tha t the sequence of measures/~1, f12 . . . . generated by distinct integers an is ergodic if and only if (an) is a so-called Hartman-uniformly distributed sequence in 7/ (cf. [2]). For information about the latter notion, we refer to [6], Ch. 4, w 5, and [7]. For the purposes of the present paper, it suffices to recall tha t (an) is Hartman-uni/ormly distributed in 7] (abbreviated: H-u.d.) if

1 N lim ~ ~ )~(an) ~- 0

~y--~ oo n = l

holds for all nontrivial characters Z of Z. Ergodic sequences of measures and H-u.d. sequences occur in a number of contexts.

Thus, a special case of a result of Hanson and Pledger [5] shows tha t the mean ergodic N

theorem holds for averages of the type .u ~ TO, / if and only if the sequence (an) n = l

of integers is H-u.d. Results of the same type are discussed in [2] and [4]. In a related development, Blum and Eisenberg [2] studied connections between ergodic sequences of measures and generalized summing sequences for almost periodic functions. The same authors showed in [3] tha t H-u.d. sequences single out those subsequences of a stationary ergodic (discrete time) process for which a law of large numbers holds.

In these considerations, certain density properties of (an) also play a role. For k ~ and a subset A of :Y, let A + k = { a + k : a ~ A } , i.e., A + k is the set A shifted by the integer k. Then it is noted in [4] tha t

(1) lim ]EN ('~ (EN + It)] N--*oo ]EN] = 1 fora l l keT/

22 H. NIEDERREITER ARCH. MATH.

is a sufficient condition for/xl , /~2, . . . to be an ergodic sequence of measures; more- over, ftl,/~2 . . . . is then a generalized summing sequence (cf. [2]). The condition (1) is, however, no t necessary. I n fact, the au thor [7] proved t h a t for any nonzero integer k and any real a with 0 --< ~r --< 1, there exists an ergodic sequence of measures ~tl,/~2 . . . . such tha t

lim I EN N (EN__+ k) I _ ~ .

Actually, the result in [7] is only s ta ted for positive k, bu t it can be extended to negative k because of Proposit ion 1 below. Also, one should note t ha t the sequence (an) constructed in the proof of [7], Theorem 2, consists o f distinct integers, so tha t the ergodicity of the associated sequence of measures is equivalent to (an) being H-u.d.

I n t ry ing to generalize the above result, Blum and Cogburn [1] claimed to have shown the following theorem. Suppose we associate with each k e 7 /a real number ~k e I ~- [0, 1] in such a way t h a t s0 ---- 1 ; then there exists an ergodic sequence of measures/xl , /~2 . . . . such tha t

lim I E`v n (EN + k) I ~V--,~ ] EN I = ~k for all k r Z .

I t will t ranspire f rom the discussion in Section 2 tha t this claim is not valid. The crucial point is t ha t the numbers ~ cannot he chosen independent ly of each other, hut are t ied together by certain identities and inequalities. Any a t t emp t to resolve the problem raised by the claim of Blum and Coghurn has to proceed hand in hand with a solution of an interesting problem in addi t ive number theory.

We acknowledge the comments of the referee which have been helpful in streamlin- ing the presentat ion of the paper.

2. Admissible tup]es. Let E --- (an) be an arbi t rary sequence of integers, and let the notat ions from Section 1 be operative. For k E 2[ we set

IENf5 (EN + k)[ Pk (E) ---- lim

~.-=,~ }EN}

whenever the limit exists. ] t is trivial tha t po(E) ---- 1 and tha t pz(E) + I = [0, 1]. We m a y th ink of Pk (E) as the , ,probabil i ty" for k occurring as a difference in the sequence E.

Definition 1. For a nonempty set K of nonzero integers, let P(/~) be the set of all r = (~tk)keK E IIKI for which there exists a sequence E of integers with pk(E) = ~ for all k e K. I f r P(K) , we say tha t Gt is an admissible (I KI ' ) tuple.

Definition 2. For a nonempty set K of nonzero integers, let PH (K) be the set of all a = (gk)kEK ~ IIKI for which there exists an H-u.d. sequence E of integers with pk(E) = ~k for all k e K. I fc t ~ PH(K), we say t h a t Gt is an H-admissible (IKI-) tuple.

Vol. 31, 1978 Additive number theory 23

I t is clear from these definitions that PH ( K ) C P ( K ) . We pose the problem of determining P ( K ) and PH(K). The relationship to the discussion in the previous section is obvious. On the basis of the information available so far, we conjecture tha t PH (K) = P (K) for all K.

We first establish links between the numbers pg {E). This can be done in a some- what more general context, by introducing the ,,lower" and ,,upper probability" pg (E) resp./bk (E) obtained from the definition of pk (E) upon replacement of ,,lim" by ,,liminf" resp. ,,limsup".

Proposition 1. We always have _Pk (E) = _p-~ (E) and ~k (E) : ~-k (E). In particular, i / T k (E) exists, then p-~ (E) exists and p-k (E) -~ Pk (E).

P r o o f . For N > 1 we have ]EN (~ (E~ § k)l = lEvy n (EN -- k)] since on both sides we count the number of pairs of elements of EN that differ by k. The results follow immediately from this identity.

Proposition 2. For any integers r and s we have

(2) Tr (E) -4- Ts(E) =< Tr+ (E) -4- 1

and

(3) /br (E) -4- _ps (E) g 15r+s (E) -4- 1.

In particular, i] pr(E), ps(E), and Pr+s (E) exist, then

(4) pr(E) -4-ps(E) < pr+s(E) -4- 1.

P r o o f . I f b ~ (EN -- r) (3 EN n (EN -4- s), then b -4- r e E~ n (EN -4- r -4- s), and so

]EN (3 (EN -4- r -4- s)] > I(EN - r) (3 E~ n (EN -4- s) I �9

By the inclusion-exclusion principle we get

](E~ -- r) t~ EN t3 (EN -4- s)] > I EN n (E~ -- r) l -4- ]EN (3 (E~ -4- s)] -- [EN ].

By combining these inequalities, dividing by I EN ] and letting N -+ r162 we arrive at

Tr+s(E) > T - r ( E ) § - 1 = T~(E) § 1,

where we used Proposition 1 in the last step. Thus (2) is shown. Furthermore, the above inequalities yield

]EN n (EN -- r)] < ]EN r (EN -4- r -4- s)] -- ]E~ t3 (EN -4- s) l -4- IEN] .

Dividing by ] Ely I and letting N -+ r162 we get

p-r(E) g pr+s(E) - -T~(E) -4- 1,

and because of Proposition 1, we can deduce (3). In particular, if p l ( E ) : 1, then it follows from Proposition 1 and a repeated

application of (2) that the pk(E) exist and are equal to 1 for all k e 2[. Clearly, the two propositions show that the result of Blum and Cogburu mentioned in Section 1 is incorrect.


Proposit ions 1 and 2 imply necessary conditions on admissible and H-admissible tuples. :For certain sets K, these conditions are also sufficient and thus lead to complete characterizations of P(K) and PH(K) (see Theorem 2 and Examples 1 and 2 below). H. L. Montgomery has shown in a pr ivate communica t ion tha t the tuple (ak) with ~r = 1/2 for ]c ~ ~ 1, 4- 3, 4- 9 and :r - - 0 for all o ther nonzero integers ]c is not admissible, so tha t for K = 7/\ {0} the conditions implied by Propo- sitions 1 and 2 are not sufficient. A. Odlyzko communica t~ l the example of the nonadmissible tuple (:ok) with :ok = 1/2 for k = 4- 1, :r = e for k = 4- 3, az ----- 5 for k = 4- 6, and ~k ~-- 0 for all other nonzero integers /c, where s :> 0 and ($ > 0 are sufficiently small. This example shows again tha t for K = 7/\ {0} Proposit ions 1 and 2 do not yield sufficient conditions, but it serves also as a counter-example to a conjecture pu t forth by Montgomery.

As far as general results are concerned, a fundamenta l geometric p roper ty of both P ( K ) and PH (K) is established by the following theorem.

Theorem 1. For any nonempty set K o] nonzero integers, P (K) and PH (K) are convex subsets o] I IKI.

Corollary 1. For any real number 2 with 0 <~ ,~ ~ 1, there exists an H-u.d. sequence E with pk (E) ~ ~ /or all integers k ~ O.

For certain finite sets K we can completely determine P ( K ) and PH(K). For tKI ----- 1, Corollary 1 implies immediately tha t P ( K ) = PH(K) = I, which was, in fact, a l ready shown in [7]. For ] K I = 2, the conditions implied by Proposi t ion 1 and 2 tu rn out to be sufficient. Because of Proposi t ion 1 we m a y assume, w.l.o.g., t ha t the two elements of K are positive.

Theorem 2. Let K : {k, m} with integers k and m satis/ying 1 ~= k ~ ra. I / k does not divide m, then P(K) = P n ( K ) = 12. I / k divides m, then

P ( K ) = PIt(K) = {(~z,am) e I 2 : r162 >= ( m / k ) ( ~ - - 1) + 1}.

On the basis of the principles involved in the proof of Theorem 2, one can show tha t for certain sets K with I K [ : 3 the conditions implied by Proposit ions 1 and 2 are again sufficient.

Example 1. Let K : - {r, s, t} with positive integers r, s, t such tha t t is divisible by both r and s, but r and s are not divisible by each other. Then

P ( K ) ~- PH(K) : {(~r,~s,~t) e I3: :~t ~ (t/r)(~r-- l) + 1,

:<~ >_ (t/s)(:<~ - l) + 1}.

Example 2. For K = {1, 2, 3) we get

P ( K ) = PH(K) = { ( ~ 1 , ~ 2 , ~ 3 ) ~ I 3 : ~ 1 + ~ 2 - - 1 < ~ a < ~ * - - ~ 2 + 1,


In the case of a set K for which Propositions 1 and 2 imply no restrictions, all possible I K l ' tuples are actually H-admissible, as shown by the following result.

Theorem 3. Let K be a set o/nonzero integers with I K I >= 2 and with each element o / K having the property that it is not divisible by the g.c.d, el the remaining elements o / K . Then P (K) ---- PH (K) = II ~rl.

We remark tha t a question analogous to tha t of determining P(K) can be posed for any abelian group and tha t Propositions 1 and 2 (with k, r, and s replaced by elements of the group) continue to hold in this general setting.

3. Proof of Theorem 1. Throughout this section, K will denote an arbi trary nonempty set of nonzero integers. The following auxiliary result is needed.

Lemma 1. For every admissible Gt = (:r there exists an unbounded sequence F el positive integers with pe(F) = ~k /or all k e K.

P r o o f . Let E = ( a n ) be a sequence of integers with pe(E) = ~r for all k e K. We construct the sequence F = (bn) as follows. First choose an integer cl such tha t bl = al ~- Cl > 0. Then choose an integer c2 such tha t b2 = al + c2 and b3 = a2 q- c2 are both ~ bl q- 1. In general, put V(r) ---- r(r Jr 1)/2 for r ~ 1 and suppose that bl, b2, . . . , by(r) have already been defined; then set

by(r)+1 = al -~ Cr+l for 1 ~-- j ~ r -]- 1 ,

where the integer Cr+l is chosen in such a way tha t

bv(r)+i~ max b n q - r for 1 ~ ] ~ r q - 1. 1 ~_n ~_ V(r)

Clearly, F is an unbounded sequence of positive integers. For N > 1 there exists r ~ l with V ( r ) < N <= V(r~- 1). By the construction

of F we have

(5) l ev i = IEjl + IEN-v(r) I , i = 1

and for any positive integer k,

(6) [Fly n (FN q- k) I = ~ [E 1 ~ (E i + k) I q- / '=1

+ IEN_v(r) C~ (EN-v(r) q- k)[ + RN(k)

with

(7) k + l i - 1

0 < R~:(k) < ~ ~. I (E~ q- c~) n (E 1 q- c I -t- k) [. i = 2 / '=1

For fixed positive k ~ K u (-- K) we get because of Proposition 1,

I E• n (EN q- k)[ lim - = ~ k ,

~v-~ IENI

2 6 H . NIEDERREITER ARCH. MATH.

from which it follows tha t

I E1 (~ (El ~-/c) I - F " " :F [Er n (Er + k)] lim IEII ~_ ... _jr iErt : : ~ k . r ---> OO

Hence, given e > 0, there exist ro(e) and N0(e) with

: lEln(E +k)l+'"+lErn(Er+k)] i l e v i =~ "'" -+- [Erl =

for all r _> ro (e) and

I EN n (EN + k) l

for all N ~ N0(e). Therefore, if N -- V(r) >--_ No(e) and r >-- ro(e), then

(8)

I E1 n (El Jr- lc)} + . . " -+ [Er n (Er Jr/c)} -+- I EN_v<r) n (EN-V(r) + k) l i < e

levi ~ - " " - + - [ E r l -t- IEN-v(r)I --:r = "

I f N -- V (r) ~, No(e) , then

I E1 t~ (El -}- k)[ + " . -+- I Er ~ (Er + ]c)] -Jr [ EN-v(r) ~ (EN-v(r) =Jr ]c)] ~_

IE,[ + "'" + [Erl -C IEy-v(r) l

IE1 o (El -~- k) l -Jr"" -~ ]Er (h (Er ~- ]c) l -F N0(e) ~ - - - - IE, I ~_ . . . _+_ iErl ~ :ok -}- e

for sufficiently large r, and also

[El r (El -+- k)] -+- --. § lEt (~ (Er ~- k) I + ] EN-V(r) n (Ez+-v<r> ~- ]c) I ___

IEl l -~ . . . ~- IErl + IEN-v(r) I I Ex n (El § k) I § 5- lEt n (Er § k) I

--> I E l l + "- + l e t [ + N0(e) _-->~ - - e

for sufficiently large r. Thus we have shown (8) for all sufficiently large N. Together with (5), (6), and (7), it follows tha t

I~'Nn (F~ + k)[ Pk (F) = lim --~ cote.

Because of Proposition 1, the proof is complete.

Corollary 2. For every admissible ot : (o~k)~+K there exists a sequence G o] distinct positive integers with Pk (G) ~ ~ /or all k e K.

P r o o f . By Lemma l, there exists an unbounded sequence F of positive integers with Pk (F) ---- a~ for all k e K. By deleting repeated elements from the sequence F, one obtains a sequence G of distinct positive integers with pk (G) ~- Pk (F) : ~k for all k e K.


To prove t h a t P (K) is a convex subset of IIKI, assume a = (:~k)k~K and g = (fl~)ke~ are admissible tuples. We have to show tha t for any given ), with 0 < ~ < 1, the tuple

+~ r + (1 - - ).) [~ ---- (~t~k -4- (1 - - +~) flg)k+K

is again admissible. By Lemma 1, there exist unbounded sequences E(i) = (an) and E (2) ---- (bn) of positive integers with pk(E(i)) ---- ak and Pk (E (2)) = fig for all k r K. For r >= 1 let S(r) and T(r) be positive integers such t h a t E (1)s(~)[ = [r/(1 - - +~)] and [ ~(2) "~T(r) ---- [r/).], where [x] denotes the integral pa r t o f a real number x. Further- more, we set S (0) ~ T (0) ~ 0. We define a sequence E = (cn) by

lan-T(r) for S(r) § T(r) < n < S(r -t- 1) § T(r ) , cn ~ ( - - bn-s(r+l) for S(r § 1) § T(r) < n < S(r ~ 1) g- T ( r - ~ 1).

P u t U(r) -~ S(r) § T(r) for r > 1. We have then

E u (r) ~ { a l , a2 . . . . . a x (r) , - - b l , - - b2 . . . . . - - bT(r)} ,

and so I Ev(r)] -~ [r/(1 - - ),)] -~ [r/+~]. I f k is a positive integer, then

E u ( r ) § k ~ - ( a 1 -}- k , a2 -1- k , . . . , a s ( r ) -4- k ,

- - bi § k, - - b2 -~ k . . . . , - - bT(r) -~- k } ,

and so the elements in Ev(r) t3 (Ev(r) -~ k) are either elements a= tha t are also of the form am -4- k, or elements - - bn t h a t are also of the form - - b m + k, or elements a , t ha t are also of the form - - b,~ A- k. The number V (r, k) of distinct elements in the lat ter ca tegory satisfies V (r, k) < k. We get

I E~(~) n (Eu(~) + k)[ = / 17(I) 17(2) ~ ( 2 ) - I E~<),) n ,=s(:) § k) l + I n § k)[ § V(r, k) - - ~:~ T(r) ~ Z~ T(r) ,

and so for positive k e K ~9 (-- K) and with the use of Proposi t ion 1,

lim ]Ev(r) n (Ev(r) § k) I

---- lira ] E!s'10r) (3 ' E (1)

E(D +E(i) ---- lira s(r) r t s(r) § k) l r~oo [ h?(1) I ~8( r ) I

--r<:) n (Er<,> + k) l . I E<~:)I § lim [~(2) (2) t h?(2)

r-+oo ~ T ( r ) I E u ( r ) l

E<I)[ ~<2) ---- ~klim I;_ -2(~) + / ~k l i m ~T(r)

provided the last two limits exist. Now

[E(~0r) l [r/(1 - - ~t)] lim . . . . = lim ----

~-~o0 Ev(~)I ~+~ [r/(i---- ~ + [~/X]

~,(2), F't 1~(2) j § l i m Im~"J_~_ ~ ' ( ~ ) ~- k)[

IEu(,)l 7 - -+ OO

E(1) s(r) IEv(r) l +

~ 8 H . N I E D E R R E I T E R A R C H . M A T H .

and

so tha t

lim [~(9=) [r/2] ~T( , ) = l i m = 1 - - 2 r+~ Eu(r) l ,~r162 [r/(1 - - 2)] + [r/2]

]Ev<, n(Ev(~) + k)} = 2~x + (1 -- 2 ) i l k . (9) r-+~o]im i Ev(r) [

For every integer N > U(1) there exists r ~ 1 with U(r) ~ N < U(r + 1). Then,

I Eu n (E~ + k) l [Eu<r) n ( E v < r ) + k ) l " [Wu(r)[ < <_

I Eu(,)I I E u ( , + , l = I E~ 'l - _-< I Ev(,+l) n (Eu(r+l) + k) I IEv(r+I) I

I Ev<,+, I {Ev<,) I '

and since

[Eu(r+i)] [(r + 1)/(1 --)~)] + [(r + 1)/2] lira - - lira ---- 1, ,_+<~ I Ev<,)l ,_+~ [r/(1 - - 2)] + [ r l~ ]

we infer f rom (9) t h a t

px(E) = lira N - - + ~

In conjunct ion admissible.

]EN n (EN + k)[

lEg i = ~ rick + (1 - - )~) fl/c.

with Proposi t ion 1, it follows t h a t the tuple 2a + ( 1 - 2 ) ~ is

L e m m a 2. For every H-admissible a = (:r there exists an H-u.d. sequence F el positive integers with Pk (F) ---- :ok ]or all k e K.

P r o o f . Le t E = (an) be an H-u.d. sequence with pk(E) = ak for all k ~ K. The sequence F ---- (bn) is now constructed in exac t ly the same way as in the proof of L e m m a 1. I t remains to show t h a t Y is again H-u.d. Le t Z be a nontr ivial charac te r of F. Using the nota t ion f rom the proof of L e m m a 1, we choose N > 1 and de termine r ~ 1 by the condition V (r) < N ~ V (r + 1). Then

N r j N - V(r)

~ : Z ( b n ) = ~ E)C(an +c l ) -~ ~ Z(an +Cr+l) = n = l j~l n = l n = ,

i A'- r(r) = x<c ) x<o.) + xIC.l l x<a.I,

j = l n = l n = l

and so

(10)

Le t d 1 - - - - I j - l ~ x ( a n ) l ;

1 .v 1 r . 1 ~ W / ~ Jtan~i r + l

f:/(b.) <: 7 + ,'(;) i = i i = J

then the sequence dl , d2, d~, da, d3, d3, . . . , with each d I


being repeated j times, tends to 0, and so the sequence of ar i thmetic means tends N

to 0 as well. Consequently, (10) implies t h a t lim N -1 ~, z(bn) -= O, and the proof is complete. ~-~oo n=l

To prove tha t PH (K) is a convex subset of I I K I, assume at = (:r K and ~ = (flk)ke ~: are H-admissible tuples. B y Lemma 2, there exist H-u.d. sequences E(1) = (an) and E(2) ---- (bn) of positive integers with pk(E(1)) = ~k and pk(E(2)) ---- flk for all k e K. For a given ;t with 0 < )~ < 1, we construct the sequence E ---- (Cn) in the same way as in the proof of the first par t of Theorem 1 (note t h a t an H-u.d. sequence is au tomat ica l ly unbounded). Then pk(E) -= ~ + (1 - - 2)flk for all k e K. To show t h a t E is again H-u.d., let g be a nontrivial character of 7 /and choose N sufficiently large. By the construct ion of E, the first N terms of E consist of an initial block of the sequence (an) of length A (N), say, and of an initial block of the sequence (-- bn) of length B ( N ) , say. Therefore,

1 N 1 A(zr 1 B(~') Zx Xx(- b.) <=

A(N) [ 1 B(N) 1 ~ z ( a n ) + ~ , z ( b n )

-<- (N5 " N

Since lira A (N) ---- lira B ( N ) = r we conclude t h a t lim N -1 ~ )~(en) = O. Thus, N---~ oo N--~oo N--+oo n = l

the tuple 2r + (1 - - 2) ~ is H-admissible, and the proof of Theorem 1 is complete.

The p r o o f o f C o r o l l a r y 1 proceeds as follows. Le t ~ = (:ok) with :ok = 1 for all integers k =4= 0 and ~ ----- (ilk) with fl~ ---- 0 for all integers k 4 0. Le t E(1) be the sequence of positive integers; then E(1) is H-u.d. and Pk (E (1)) = ~z for all k # 0. Choose a > 1 with a ~ 2[ and let E(e) be the sequence E(U) = ([na]), n ---- 1, 2 . . . . . of integral par ts o f n a. F rom lim ([(n + 1) a] - - [na]) -~ oo it follows t h a t p~(E(e)) -~ flk

for all k 4= 0. Fur thermore , E(2) is H-u.d. by [7], Corollary 3. Therefore, both a and are H-admissible. The desired result is now implied by the second pa r t of Theorem 1. We note t h a t the sequence of measures/~i, /~2 . . . . associated with the sequence E

constructed in Corollary 1 is ergodic since, by the construct ion in the proof of Theorem l, the sequence E consists of distinct integers.

4. Proof of Theorem 2. The following auxiliary result will be useful.

Lemma 3. Given a positive integer h, there exists an H-u. d. sequence E such that p~ (E) = 1 /or all i divisible by h and p~ (E) = 0 / o r all i not divisible by h.

P r o o f . Le t (an) be an H-u.d. sequence with a,+l -- an > hn for all n > 1. Such a sequence exists: take, for instance, a sequence ([na]), n = 1, 2 . . . . . with a sufficiently large a ~ 7]. Now the sequence E ---- (bn) is in t roduced as follows:

al , a2, a2 + h, aa, aa + h, a3 + 2h . . . . .

Thus, the sequence E is made up of blocks, with the ]-th block consisting of the j terms aj, aj -f- h . . . . . al A- (] -- 1) h. I t is then s t ra ightforward to check tha t the

30 H . NIEDERREITER ARCH. MATH.

p~ (E) have the desired values. To show t h a t E is H-u.d., let Z be a nontrivial character of )7, choose N :> 1, and determine r ~ 1 by the condition V(r) < N ~ V(r § 1), where V(r) = r(r + 1)/2. By the construct ion of E, we have

1 x 1 r i - a 1 N

n = l j = = N n = v ( r ) + l

r i-i 1 N 1 ~_lz(a,)n~=o(Z(h))n § ~ g(bn) �9 Ni= = - - V ( r ) + l

I f g(h) ---- 1, then

1 r . r + l ] 1 ~v i Vir~_j ~ = (b.) < ~z(aJ) + V ( r ) '

and using summat ion by parts,

r r i r ~ j x(at) ---- -- ~, ~X(an) § (r § l) ~,X(an ). j=l i = l n = l n = l

Therefore,

i r-n~--iz(an) + - - - - ' Nn~=l Z(bn) ~--V~i~=i ? 7n~=? (an) + V(r) = V(r)

2r and by the a rgument following (10) we obtain lim N-i~g(bn) --- O. I f x(h) # 1, then /v-.oo n=l

i ~v i ~ z(ih)--i i N - Zz(b-) = lz(a,) + Z z(b.),

a n d SO

= (bn) ~ V ( r ) [ z ( h ) _ l l § V(r----)'

N

which implies again lira N -1 ~ Z (bn) ---- O. ~V--* oo n ~ l

:Now let K ---- (It, m} be as in Theorem 2. I f k does not divide m, it suffices to show t h a t PH(K) = 12. Since PH(K) is a convex subset of 12, we need only prove tha t the four pairs (0, 0), (1, 1), (0, 1), and (1, 0) are H-admissible. For the first two pairs, one m a y either use L e m m a 3 or Corollary 1, and for the last two pairs one uses Lemma 3 with h ---- m and h ---- k, respectively.

I f k divides m, then a repeated application of (2) shows t h a t the condit ion

. m >= ( r e ~ k ) ( ~ - - l) + 1

is necessary for an admissible pair (:r ~m). Therefore, it remains to prove tha t every (~k, :r e 12 with am ~ (m/k)(:r -- 1) + 1 is H-admissible. This inequal i ty deter- mines a trapezoid with vertices (0, 0), (0, 1), (1, 1), and (1 - - (k/m), 0). Since PH(K) is convex, it suffices to show tha t these four pairs are H-admissible. The fact t h a t the


first t h ree pa i rs are H-admiss ib le is es tab l i shed in the same w a y as above. To show t h a t the las t pa i r is H-admiss ib le , let (an) be an H-u .d . sequence with

l im (an+i - - an) = r

(e.g., the sequence E(2) in the p roof of Corol lary 1), and consider the sequence

E = (bn) cons t ruc ted as follows:

al ,az + k . . . . . al + - - 1 k ,

a2,a2 + k . . . . . a2 + ( ~ - - l ) k , . . . ,

I t is seen i m m e d i a t e l y t h a t Pk (E) = 1 - - (k/m) and Pm (E) = 0. As to E being I t -u .d . , let X be a non t r iv ia l cha rac te r of 7 / a n d wr i te a g iven N > m/k in the form N = q(m/k) + r with q => 1 and 0 ~ r < talk. Then,

1 iv 1 q (mlk)--I m

N ~ I ~ ~ j= z(ai + i k) + N k --

= =~i g(aJ) z--~o x(i]c) + < z(at) + q

2v and so l im N -1 ~. g (bn) = O.

. N -..>. o ~ n = l

5. P r o o f of Theorem 3. Le t K be as in Theorem 3. Wo show t h a t all tup les (e~)k~K, wi th t he ek t ak ing i ndependen t ly t he values 0 and 1, a re H-admiss ib le . F o r the zero tuple , th is follows from Corol la ry 1. I f (~)~eK is one o f the remain ing tuples , le t J = ( j e K : ~1 = 1}. W e a p p l y L e m m a 3 with h being the g.e.d, of al l e lements of J and t h e r e b y ge t an H-u .d . sequence E with Pl (E) = 1 - - ~j for all j e J . F o r k ~ K \ J , i t follows f rom the hypo thes i s on K t h a t h does no t d iv ide k, and so pk(E) = 0 = ek b y L e m m a 3. Therefore, (ek)k~K is an H-admiss ib le tuple .

References

[1] J. R. BLUY[ and R. COOBURN, On ergodic sequences of measures. Prec. Amer. Math. Soc. 51, 359--365 (1975).

[2] J. R. BLVM and B. EISE~BERG, Generalized summing sequences and the mean ergodie theorem. Prec. Amer. Math. Soc. 42, 423--429 (1974).

[3] J . R. BLUM and B. EISENBERG, The law of large numbers for subsequences of a stationary process. Ann. of Prob. 3, 281--288 (1975).

[4] J. R. BI,UM, B. EISENBERG, and L.-S. HAHN, Ergodic theory and the measure of sets in the Bohr group. Acta Sci. Math. (Szeged) 34, 17--24 {1973}.

[5] D. L. ttA~SON and G. I~EDGER, On the mean ergodic theorem for weighted averages. Z. Wahrscheinliehkeitsth. verw. Geb. 13, 141--149 (1969).


[6] L. KUrPERS and H. NIEDERR:EITER, Uniform distribution of sequences. New York 1974. [7] H. NIEDERREITER, On a paper of Blum, Eisenberg, and Hahn concerning ergodic theory and

the distribution of sequences in the Bohr group. Acta Sci. Math. (Szeged) 87, 103--108 (1975).

Anschrift des Autors:

Harald Niederreiter Department of Mathematics University of Illinois Urbana, IL 61801, U.S.A.

Eingegangen am 25. 10. 1976 *)

Ab 15.8. 1978:

Chair in Pure Mathematics University of the West Indies Kingston 7 Jamaica

*) Eine iiberarbeitete Fassung ging am 9. 5. 1978 ein.

Documents

Ergodic sequences of measures and a problem in additive number theory