Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

LÁSZLÓ , B . & T. TÓTH, J., On very porosity and spaces of generalized uniformly distributed sequences

56 Béla László & János T. Tóth holds (cf. [3], p. 7). Denote U = {x = (,x

nG 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-cporous 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.