Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) ON VERY POROSITY AND SPACES OF GENERALIZED UNIFORMLY DISTRIBUTED SEQUENCES

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1 Acta Acad. Paed. Agriensis, Sectio Mathematicae ) 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 J7, Secondary K36, B05 Keywords: Uniform distribution, Baire space, porosity. 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 [], 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 S in the following way. Let x = x n ) S, y = y n ) S. If x = y, then dx, y) = 0 and if x y, then dx, y) = min{n : x n 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 = x n ) s is uniformly distributed and only if for each integer h 0 the equality mod ) if lim N N N e 2πihxn = 0 n= This research was supported by the Czech Academy of Sciences GAAV A 87 0.

2 56 Béla László & János T. Tóth holds cf. [3], p. 7). Denote hence from Theorem A we have U = { x = x n ) U = {x = x n ) s; x n) is u. d. mod }, s; lim N N } N e 2πihxn = 0 for each integers h 0. n= We now give definitions and notation from the theory of porosity of sets cf. [5]-[7]). Let Y, ) be a metric space. If y Y and r > 0, then denote by By, r) the ball with center y and radius r, i.e. Let M Y. Put By, r) = {x Y : x, y) < r}. γy, r, M) = sup{t > 0 : z Y [Bz, t) By, r)] [Bz, t) M = ]}. Define the numbers: py, M) = lim r 0 + sup γy, r, M), py, M) = lim inf r r 0 + γy, r, M). r Obviously the numbers py, M), py, M) belong to the interval [0, ]. A set M Y is said to be porous c-porous) at y Y provided that py, M) > 0 py, M) c > 0). A set M Y is said to be σ-porous σ-c-porous) at y Y if M = M n and each of the sets M n n =, 2,...) is porous c-porous) at y. n= Let Y 0 Y. A set M Y is said to be porous, c-porous, σ-porous and σ-cporous in Y 0 if it is porous, c-porous, σ-porous and σ-c-porous at each point y Y 0, respectively. If M is c-porous and σ-c-porous at y, then it is porous and σ-porous at y, respectively. Every set M Y which is porous in Y is non-dense in Y. Therefore every set M Y which is σ-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 Y is said to be very porous at y Y if py, M) > 0 and very strongly porous at y Y if py, M) = cf. [7] p. 327). A set M is said to be very strongly) porous in Y 0 Y if it is very strongly) porous at each y 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 -porous at y.

3 On very porosity and spaces of generalized uniformly distributed sequences 57 Further, a set M Y is said to be uniformly very porous in Y 0 Y provided that there is a c > 0 such that for each y Y 0 we have py, M) c cf. [7], p. 327). In agreement with the previous terminology and in analogy with the notion of σ porosity, we introduce the following notions. A set M Y is said to be uniformly σ very porous in Y 0 Y provided that M = M n and there is a c > 0 such n= that for each y Y 0 and each n =, 2,... we have py, M n ) c. 2. Main Result In this part of the paper we shall study the set of all uniformly distributed mod ) sequences in the space s, d). Evidently for an integer h > 0 we have U S h) = { x = x n ) s; lim N N } N e 2πihxn = 0 n= r= n=r Fk, n) for every k =, 2,..., where Fk, n) = x = x n) s; n n j= e 2πihxj k. Denote F k, r) = Fk, n) for k =, 2,...,r =, 2,... n=r First, for f : R R let us denote S h) f) = x = x n) s; lim n n n e 2πihfxn) = 0 j= and similarly Uf) = {x = x n ) s; fx n )) is u. d. mod }. The next theorem implies, that the set S h) is σ very porous in s, d). Hence, it follows that σ very porous in U too, see Corollary 2.) Theorem. Let f : R R be a function. Then the set S h) f) is uniformly σ very porous in s, d).

4 58 Béla László & János T. Tóth Proof. For f : R R and k =, 2,... denote by Then we have Let Ff, k, n) = x = x n) s; S h) f) F f, k, r) = r= n=r n n j= Ff, k, n). Ff, k, n). n=r e 2πihfxj) k. Choose r N fixed. Let ε > 0 and x s. Further let δ > 0 be such that δ < r. Then there exists a positive integer l such that l δ < l, consequently l > r). Obviously S h) f) F f, 2 + [ ) 3, r. Therefore it suffices to prove r= p x, F f, 2 + Choose a sequence y s as follows: [ ] )) 3, r ε 2. { xj, for j =, 2,..., l, y j = b, for j > l, where b is constant. Evidently y B x, l will show B y, Let z B y, [2+ε)l]+ > [2 + ε)l] + [2 + ε)l] + [2 + ε)l] + ). Then we have [2+ε)l]+ j= l j= e 2πihfzj) e 2πihfzj) ) ) and B y, [2+ε)l]+ B ) x, l. We ) F f, 2 + [2 + ε)l] ε)l 2l [2 + ε)l] + εl 2 + ε)l + = ε 2 + ε + l [2 + ε)l] + l [2 + ε)l] + [ ] ) 3, r =. ε [2+ε)l]+ j=l+ e 2πihfzj) [2 + ε)l] + > ε ε + 3 = 3 ε + > [ 3, + 2

5 On very porosity and spaces of generalized uniformly distributed sequences 59 thus z / F f, 2 + [ 3, r). Then γx, δ, F f, 2 + [ 3, r)) δ [2+ε)l]+ l l 2 + ε)l +, i.e.) p x, F f, 2 + [ ] )) 3, r ε 2 + ε and letting ε 0 we obtain the required inequality. Remark. Since the set F f, 2 + [ 3, r) is closed in s, for each x s\f f, 2 + [ 3, r ) holds p x, F f, 2 + [ ] )) 3, r =. ε Corollary. Let f : R R be a function. Then the set Uf) is uniformly σ very porous in s, d). Corollary 2. The set S h) is uniformly σ very porous in s, d) for every h positive integers. Proof. It follows from the fact that the function fx) = x, x R. References [] Kuipers Niederreiter, H., Uniform distribution of sequences, Wiley, New York, 974). [2] László, V. Šalát, T., Uniformly distributed sequences of positive integers in Baire s space, Math. Slovaca 4. no 3 99), [3] László, V. Šalát, T., The structure of some sequences spaces, and uniform distribution mod ), Periodica Math. Hung., Vol. 0) 979), [4] Niven, I., Uniform distribution of sequences of integers, Trans. Amer. Math. Soc., 98 96), [5] Tkadlec, J., Construction of some non σ porous sets of real line, Real Anal. Exch., ), [6] Zajíček, L., Sets of σ porosity and sets of σ porosity q), Čas. pěst. mat., 0 976),

6 60 Béla László & János T. Tóth [7] Zajíček, L., Porosity and σ porosity, Real Anal. Exch., ), Béla László Department of Algebra and Number Theory Constantine Philosopher University Tr. A. Hlinku Nitra, Slovakia blaszlo@ukf.sk János T. Tóth Department of Mathematics Faculty of Sciences University of Ostrava 30. Dubna, Ostrava Czech Republik toth@osu.cz