*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 = (y**n**)i° G S. If x = y, then d(x,y) = 0 and if x ^ y, *
then

*m m { n : x**u**, f . y**n**) *

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 = ( , x** ^{n}*G s; (a?

^{n})f is u. d. mod 1},

*hence from Theorem A we have*

*U = [x = (®„)°° G .s; lim V e*^{2,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 M*^{n}* and each of the sets M*^{n}* (n = 1 , 2 , . . . ) is porous (c-porous) at y. *

*n-l *

Let Y^{0}* 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 M**n** and there is a c > 0 such *
n=l

*that for each y E YQ and each n = 1 , 2 , . . . we have p(y, M**n*) > 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 = (x**n**)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

^{ e}

^{2*ihJ{x}

^{n}

^{)}

^{ =}

^{ Q }*n —> m 11 < * *
i = l
and similarly

*U ( f ) = {x = (x*n*)5° E s; (f(x**n**j)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(f*^{t}*k,n)= {x = (x*^{n}*)™ 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,k*^{t}*r)= 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