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

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 55-60 ON VERY POROSITY AND SPACES OF GENERALIZED UNIFORMLY DISTRIBUTED SEQUENCES Béla László (Nitra, Slovakia) &: János T. Tóth (Ostrava, Czech Rep.) Abstract. In the paper the porosity structure of sets of generalized uniformly distributed sequences is investigated in the Baire's space. AMS Classification Number: Primary 11J71, Secondary 11K36, 11B05 Keywords: Uniform distribution, Baire space, porosity 1. Introduction and definitions 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

nG S, y = (y

n)i° G S. If x = y, then d(x,y

) = 0 and if x ^ y, then mm

{n : x

u, f. y

n) In [2] is proved that 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: Theorem 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.