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

58 Béla László & János T. Tóth Proof. For / : R —* R and k = 1,2,... denote by F(f

tk,n)= {x = (x

n)™ Es1 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

tr)= 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 < py, (consequently I > r). CO Obviously S^{f) C [jr (/,2+ [f] ,r). Therefore it suffices to prove p[x,F* f, 2 + Choose a sequence y £ s as follows: _ , Xj, for j = 1,2, ~ 1 b, for 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+

g

1

);]

+

1 j - Then we have n F* /, 2 + ,r = [(2+c)I]+l [(2 + e)/] + 1 E i =

1 ,2 TTihf(Zj) > R2 + 0U + 1 [(2+00+1 I Y ,2irihf(zj) > [(2 + e)l] + 1 — / _ 1 [(2 + e)l] -f 1 [(2+ £)/] + ! (2 + £•)/ — 2

1 > [(2+ e)/]+l - (2 + e)/ + l 2 + s+t > - + 3 F + 1 ^ [?]+

2' 1 1 >