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

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 <Jporosity, 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. Main Result 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 rrln=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; li

m I y e2*ihJ{x

n) = Q n —> m 11 < * i = l and similarly U(f) = {x = (x

n)5° E s; (f(x

nj)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.) Theorem. Let f : R —> R be a function. Then the set S

(h

)(f) is uniformly a-very porous in (s, c/).