Porosity of the Two Ration Cantor Set
by Priyanka*,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 1, Issue No. 1, Feb 2011, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
Inthis note we prove that the two ration Cantor set is uniformly very porous inthe set of real numbers.
KEYWORD
porosity, two ration Cantor set, uniformly very porous, real numbers
1. Introduction
Let r1 and r2 be positive real numbers such that r1+r2 <1. Put and so on, For any positive integer k, Ek is a union of 2k disjoint closed intervals, If [a,b] is a typical interval in Ek then the intervals in Ek+1 are given by [r1a,r1b] and [r1a,r1b]. Further if c be such that r1
2. Some Basic facts about Porosity
We recall the following basic facts about porosity. Let RMand let y be a real number. Let suchthat and Note that if M is dense then Let and if then we set The set M is said to be porous at y if p(y,M)>0 and porous at y provided Mn and each of the sets Mn is porous at y. M is called porous or porous set if it is so at each A porous set is nowhere dense while a set is a set of first category. Very simple examples of porous sets are the set of natural numbers and the set of all integers as also the classical Cantor set. The set of all retional numbers is an example of a set which is not porous. However not all nowhere dense set are porous and there are example of set which are nowhere dense but not porous. The set M is said to be very porous at y if p(y,M) > 0 and M is said to be uniformly very porous in if there is a c >0 such that for eaxh we have A very porous set or a uniformly very porous set is evidently porous but the converse is not generally true. As already mentioned, for the classical Cantor set C, it is known that C is porous at each point of C. We prove here a stronger result for C(r1,r2) in respect of porosity. Incidentally, on taking our theorem extends the theorem on porosity of C.
3. Theorem
Theorem 3.1. The set C(r1,r2) is uniformly very porous in R. Proof : if, then clearly So let then for allnon- negative integers k. Let us take I0 = [0,1] and t0 = length(I0) = 1. Since Ek is the union of disjoint closed intervals, there exists such an interval Ik from Ek that kIx and this is true for each k Thus {Ik} is a strictly decreasing sequence of closed intervals with length(Ik) = tk (say) tending to zero as k. Let r > 0. Without any loss of generality we choose r <1. Select a positive integer m such that. . it follows that tm+1 = r1tm or tm+1 = r2tm . Now since and r>tm+l = length(Im+l), we must have The Interval Im+1 contains an interval of length (1-rl-r2)tm+l which does not contain any point of C(r1,r2). Choose Then say. Thus c is positive and This shows that C(r1,r2) is uniformly very porous in R and thus the theorem is proved.
References
W.A. Coppel. An interesting Cantor set, Amer. Math. Monthly, (1983), 456-460. T. Salat, S.James Taylor and J.T.Toth, Radii of Convergence of power series, Real Analysis Exchange, 24(1), (1998-99), 263-274. L. Zajicek, Porosity and, Real Anal. Exchange, 13, (1987-88), 314-350.
