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AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 (2006) A Different Description of a Family of Middle-α Cantor Sets Mohsen Soltanifar1 Department Of Mathematics Faculty of Science K.N. Toosi University of Technology P.O. Box 16315-1618 Tehran, IRAN Received: June 17, 2006 Accepted: August 3, 2006 ABSTRACT In this note we give an explicit description of a family of middle-α Cantor sets, with α = (q – 2)/q and q = 3, 4, 5… I. INTRODUCTION Georg Cantor (1845-1918), the founder of axiomatic set theory studied many interesting sets. He was very interested in infinite sets, in particular those with strange properties. One of the sets that he constructed is named after him: the middle-third Cantor set. The middle third Cantor set C is the set of all x ∈ [0, 1] with the ternary expansion ∑∞==13nnnax, with an = 0, 2 for all n∈N. This set and its construction is an interesting and amazing set for any sophomore student in analysis. It is uncountable, perfect, compact, and has some other nice properties which makes it a curious set. On the other hand, the construction of the set shows that the role of the middle third of the interval can be replaced by any α ∈ (0, 1), and a straightforward generalization of this set, the middle-α Cantor set can be obtained [1]. To define a middle-α Cantor set in the interval [0, 1], let α ∈ (0, 1) and β = (1 − α)/2, then the middle-α Cantor set Γα is the set of all x ∈ [0, 1] with expansion ,10∑∞=−=nnnaxββ And an = 0,1 for all n ∈ ℕ0. We notice that when α = β = ⅓ this representation is equivalent to the representation of points in the middle third Cantor set. By using the idea of affine maps we can have an equivalent definition of the Γα [1]. For this, with the same assumptions on α and β, define the maps xxTβ=0 and (ββ−+=11xxT Now let I0 = [0, 1] and for n ≥ 1, define In inductively as follows: .1110−−ΙΙ=ΙnnnTTU Since T0 and T1 are continuous and closed, they take each closed subinterval of [0, 1] into a closed subinterval of [0, 1], and each In is a disjoint union of 2n closed subintervals of [0, 1] which are the image of 2n−1 closed subintervals of [0, 1] under T0, T1. To illustrate how these intervals are constructed, let’s look at them more closely. By the definition of T0 and T1, T0(I0) = [0, β] and T1(I0) = [1 − β,1], respectively, and by the definition of I1, 1 Email: soltanifar@sina.kntu.ac.ir 9 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 (2006) [][.1,1,001001ββ−=ΙΙ=ΙUUTT The removed interval has length 1 − 2β = α. In the same way, [][]ββββ,1,0210−=ΙUT and [][],,111,1211βββββββ−+−−+−=ΙUT each of four components has length β2, and the length of the removed interval from each component of I1 is β(1−2 β) = αβ, hence [][][][].1,11,1,,0222211102ββββββββ−+−−−=ΙΙ=ΙUUUUTT In general, each map T0, T1 sends 2n−1 disjoint closed intervals into 2n−1 disjoint closed intervals, hence their union, which is In, is the disjoint union of 2n disjoint closed intervals of length βn. We get In by removing from the middle of each component of In -1 an open interval of length αβn−1. By the construction of In, K⊃Ι⊃Ι⊃Ι⊃Ι3210 and each In is a closed, and hence compact, subset of [0, 1]. This collection has the finite intersection property so again by the compactness of [0, 1] they have a nonempty intersection. Then the middle-α Cantor set is defined as: nnΙ=Γ∞=0Iα. II. THE MAIN RESULT The following theorem gives an explicit description of a family of Γαs. Theorem: Let Γα be a middle-α Cantor set. Then []mmmqqkqqkqqkm11101,1,01−++−=∞=−−=ΓUUα for qq2−=α and q = 3, 4, 5… To prove this theorem, we need the following two lemmas. Before stating them and their proofs let nnΙ−Ι=Ι0* for n = 1, 2… a. Lemma 1 With the above definitions and notations, the equality *11*1*10*−−ΙΙΙ=ΙnnnTTUU holds for every n ≥ 1. Proof: By the definition of : *nΙ .*10*1*11*11*10010*10*11000010000*11010*10000*11010*10000*1010*1000110100111000*−−−−−−−−−−−−−−−−ΙΙΙ=ΙΙΙ−ΙΙΙΙ−ΙΙ−ΙΙ−Ι=ΙΙ−ΙΙΙ−Ι=Ι−Ι−ΙΙ−Ι−Ι=Ι−Ι−ΙΙ−Ι−Ι=Ι−ΙΙ−Ι=ΙΙ−Ι=Ι−Ι=ΙnnnnnnnnnnnnnnnnnnTTTTTTTTTTTTTTTTTTTTTTTTUUIUIUIUIUIUIIIU 10 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 (2006) b. Lemma 2 With the above definitions and notations, the equality mmmqqkqqkqqknmn11101*,1−++−==−=ΙUU holds for every n ≥ 1. Proof. We prove the assertion by induction. It is clear that it holds for n = 1. Let it hold for positive integer n−1, then Lemma 1 and the inclusion qqqqqkqqkqqqqknmmmmmm1111111,,1111−−++−−=−=⊆++−−UU imply that: .,,,,,,,,,,1110111101111111111111111110111111111111011*11*1*10*11111111111111111111mmmmmmmmmmmmmkmmmmmmmmmmmmmmqqkqqkqqknmqqkqqkqqknmqqqqqkqqkqqqqknmqqkqqqqqqknmqqqqqkqqkqqknmqqkqqkqqqqknmqqqqkqqkqqknmnnnTT−++−==−++−=−=−−++−−=−=−+−−=−=−−++−=−=−++−−=−=−+++−=−=−−−++++−+++−−++−++−++−===∪=ΙΙΙ=ΙUUUUUUUUUUUUUUUUUUUUU Now by virtue of Lemma 2, we prove the theorem: (mmmqqkqqkqqkmnnnnnnnn111010*10*00*000,1−++−=∞=∞=∞=∞=∞=−−Ι=Ι−Ι=Ι−Ι=Ι−Ι=Ι=ΓUUUUIIα for every qq2−=α, K5,4,3=qc. Corollary The above theorem gives the ordinary Cantor set. Proof. It is sufficient to put q =3: [].,1,03233131301131mmmkkkm++−=∞=−−=ΓUU d. Question As we mentioned in the introduction, the middle-α Cantor set can be defined for any α ∈ (0, 1) and even more for any closed interval [a, b], a < b. In this note, we obtained an explicit formula for these sets, for qq2−=α and q = 3, 4, 5… Now a natural question to raise is: Can we construct these sets whenever qpq2−=α with (p, q) = 1, q − 2p > 0 and p, q ∈ {3, 4, 5, · · ·}? REFERENCE 1. R. Kraft. “What’s the difference between Cantor sets?” Amer. Math. Monthly 101(1994), no.7, 640-650. K.N. Toosi University of Technology is one of the largest academic and technological institution in Tehran and one of the best known throughout Iran. In all its programs, undergraduate and graduate, K.N. Toosi University of Technology endeavors to provide an environment to promote a high level of professional preparation and performance for students from various backgrounds and areas of the country. It also promotes inter-academic relations with distinguished home and foreign institutions on the basis of exchange agreements. http://www.kntu.ac.ir/ 11 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 (2006) 12