Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 55-60
O N V E R Y P O R O S I T Y A N D S P A C E S O F G E N E R A L I Z E D U N I F O R M L Y D I S T R I B U T E D S E Q U E N C E S
B é l a L á s z l ó ( N i t r a , S l o v a k i a ) &: J á n o s T . T ó t h ( O s t r a v a , C z e c h R e p . )
A b s t r a c t . In the paper the porosity structure of sets of generalized uniformly distributed sequences is investigated in the Baire's space.
A M S Classification N u m b e r : Primary 11J71, Secondary 11K36, 11B05 Keywords: Uniform distribution, Baire space, porosity
1. I n t r o d u c t i o n a n d d e f i n i t i o n s
In [4] the concept of uniformly distributed sequences of positive integers mod m (m > 2) and uniformly distributed sequences of positive integers in Z is introduced (see also [1], p. 305).
We recall the notion of Baire's space S of all sequences of positive integers.
This means the metric space S endowed with the metric d defined on S x S in the following way.
Let x — ( x n G S, y = (yn)i° G S. If x = y, then d(x,y) = 0 and if x ^ y, then
m m { n : xu, f . yn)
In [2] is proved t h a t the set of all uniformly distributed sequences of positive integers is a set of the first Baire category in (S,d). In the present paper we shall generalize this result to the space of all real sequences.
Denote by (s, d) the metric space of all sequences of real numbers with d Baire's metric.
In the sequel we use the following well-known result of H . Weyl:
T h e o r e m A . The sequence x = G s is uniformly distributed (mod 1) if and only if for each integer h ^ 0 the equality
1 N
lim 1 y = 0 n = 1
This reseaich was supported by the Czech Academy of Sciences GAAV A 1187 101.
56 Béla László & János T . Tóth
holds (cf. [3], p. 7).
Denote
U = {x = ( , xnG s; (a?n)f is u. d. mod 1}, hence from Theorem A we have
U = [x = (®„)°° G .s; lim V e2,riAa:» = 0 for each integers h / 0 1 .
I n=l J We now give definitions and notation from the theory of porosity of sets (cf.
[5]-[7]). Let (Y, g) be a metric space. If y G Y and r > 0, then denote by B(y,r) the ball with center y and radius r, i.e.
B{y,r) = {x G Y : g(x,y) < r}.
Let M C Y. Put
j(y,r,M) = sup{t > 0 : 3 ,€y [B(z,t) C B(y, r)] A [B(z, t) n M = 0]}.
Define the numbers:
p(y,M)= lim sup —dl—1—p(y,M)= lim inf —-——.
7--1-0+ r — ' r—o+ r
Obviously the numbers p{y,M), p(y,M) belong to the interval [0, 1].
A set M C Y is said to be porous (c-porous) at y G Y provided that p(y, M) >
0 (p(y, M) > c > 0). A set M C Y is said to be cr-porous (cr-c-porous) at y G Y if CO
M — (J Mn and each of the sets Mn (n = 1 , 2 , . . . ) is porous (c-porous) at y.
n-l
Let Y0 Q Y. A set M C Y is said to be porous, c-porous, cr-porous and cr-c- porous in Yo if it is porous, c-porous, cr-porous and cr-c-porous at each point y G Yo, respectively.
If M is c-porous and cr-c-porous at y, then it is porous and cr-porous at y, respectively.
Every set M C Y which is porous in Y is non-dense in Y. Therefore every set M C Y which is cr-porous in Y, is a set of the first category in Y. The converse is not true even in R (cf. [6]).
A set M C Y is said to be very porous at y G Y if p(y, M) > 0 and very strongly porous at y G Y if p(y, M) = 1 (cf. [7] p. 327). A set M is said to be very (strongly) porous in Yo C Y if it is very (strongly) porous at each y G Y.
Obviously, if M is very porous at y, it is porous at y, as well. Analogously, if M is very strongly porous at y, it is 1-porous at y.
On very porosity and spaces of generalized uniformly distributed sequences 57 Further, a set M C Y is said to be uniformly very porous in YQ C Y provided that there is a c > 0 such that for each y E YQ we have p(y, M) > c (cf. [7], p. 327).
In agreement with the previous terminology and in analogy with the notion of <J- porosity, we introduce the following notions. A set M C Y is said to be uniformly
oo
(x-very porous in ?o C Y provided that M — \J Mn and there is a c > 0 such n=l
that for each y E YQ and each n = 1 , 2 , . . . we have p(y, Mn) > c.
2. M a i n R e s u l t
In this part of the paper we shall study the set of all uniformly distributed (mod 1) sequences in the space (s,d).
Evidently for an integer h > 0 we have
/Y ^ oo co
x- = (x„)f E 5; liiii - E = 0 C (J H F(k, n)
?i = l J r r l n = r for every k = 1 , 2 , . . . , where
F(k,n)= {x = (xn)T Es;
n '
j= 1
2irihx, 1
< - - k
Denote
F*(k, r) = p | F(k, n) for k = 1, 2 , . . . , r = 1, 2 , . . .
n =r First, for / : R — R let us denote
S<A)(/) = )X = ( a ;n) ~ G s; lim
I y
e2*ihJ{xn) = Qn —> m 11 < * i = l and similarly
U ( f ) = {x = (xn)5° E s; (f(xnj)T is u. d. mod 1}
The next theorem implies, that the set S"1' is cr-very porous in (s,d). (Hence, it follows that cr-very porous in U too, see Corollary 2.)
T h e o r e m . Let f : R —> R be a function. Then the set S( h )( f ) is uniformly a-very porous in (s, c/).
58 Béla László & János T. T ó t h
P r o o f . For / : R —* R and k = 1 , 2 , . . . denote by
F(ftk,n)= {x = (xn)™ Es-
1 n
- T , n L—' i = i
2wihJ(xj) - k
Then we have
Let
S{ h )( f ) C U f ] F ( f , k , n ) .
F*(f,ktr)= f | F(f,k,n).
n=r
Choose r £ N fixed. Let £ > 0 and x E s. Further let 6 > 0 be such that 6 <
r
Then there exists a positive integer / such that j < 6 < p y , (consequently I > r).
CO
Obviously S ^ { f ) C [ j r ( / , 2 + [ f ] , r ) . Therefore it suffices to prove
p[x,F* f , 2 +
Choose a sequence y £ s as follows:
_ , X j , f o r j = 1,2,
~ 1 b , f o r j > /,
where 6 is constant. Evidently y £ B (x, j ) and B i^y, j C B (x, j ) . We will show
B y, 1
[(2 + e)I\ + 1 Let z £ B iy, [(2+g1);]+1 j - Then we have
n F * / , 2 + , r =
[ ( 2 + c ) I ] + l
[(2 + e)/] + 1
E
i =1
,2 TTihf(Zj) >
R2 + 0 U + 1
[(2+00+1
I Y ,2irihf(zj) > [(2 + e)l] + 1 — / _ 1
[(2 + e)l] -f 1 [(2+ £)/] + !
( 2 + £•)/ — 21 >
[ ( 2 + e ) / ] + l - ( 2 + e ) / + l 2 + s + t > - + 3 F + 1 ^ [ ? ] +2'
1 1 >
O n very porosity a n d s p a c e s of generalized uniformly d i s t r i b u t e d sequences 59
thus z <£ F*(f, 2 H- [ —I , r). Then
j ( x , ó , F * ( f , 2 + [ g ] , r ) ) > [(2+7)11+1 > / - 1 l
/ - 1 (2 + e ) / + l
p[x,F*[f, 2 + r >
2 + e and letting e —» Ü we obtain the required inequality.
R e m a r k . Since the set F*(f, 2 + [p] , r) is closed in s, for each x £ s\F* ( / , 2 + [ f ] , r ) holds
P x, F / , 2 + r = 1.
C o r o l l a r y 1. Let f : R —»• R be a function. Then the set U(f) is uniformly cr-very porous in (s, d).
C o r o l l a r y 2. The set S^ is uniformly a -very porous in (s, d) for every h positive integers.
P r o o f . It follows from the fact that the function f(x) = x, x £ R.
R e f e r e n c e s
[1] KUIPERS - N I E D E R R E I T E R , H . , U n i f o r m d i s t r i b u t i o n of s e q u e n c e s , W i l e y , N e w
York, (1974).
[2] LÁSZLÓ, V. - .S'ALÁT, T . , Uniformly distributed sequences of positive integers in Baire's space, Math. Slovaca 41. n o 3 (1991), 277 - 281.
[3] LÁSZLÓ, V . - ,S'ALÁT, T . , T h e s t r u c t u r e of s o m e s e q u e n c e s s p a c e s , a n d
uniform distribution (mod 1), Periodica Math. Hung., Vol. 10(1) (1979), 89 - 98.
[4] NIVEN, I., Uniform distribution of sequences of integers, Trans. Amer. Math.
Soc., 98 (1961), 52 - 61.
[5] TKADLEC, J., Construction of some non-cr-porous sets of real line, Real Anal.
Exch., 9 (1983 - 84), 473 - 482.
[6] ZAJÍcEK, L., Sets of cr-porosity and sets of <r-porosity (q), Cas. pest, mat., 101 (1976), 350 - 359.
60 Béla László & János T. Tóth
[7] ZAJÍCEK, L., Porosity and ÍI-porosity, Real Anal. Exch., 13 (1987 - 88), 314 - 350.
B é l a L á s z l ó
Department of Algebra and Number Theory Constantine Philosopher University
Ti-. A. Hlinku 1.
949 74 Nitra, Slovakia e-mail: blaszlo@ukf.sk
J á n o s T . T ó t h
Department of' Mathematics Faculty of Sciences
University of Ostrava 30. Dubna, 22 70 103 Ostrava Czech Republik e-mail: toth@osu.cz
