MathDB

For

A\subset\mathbb Z

and

a,b\in\mathbb Z

. We define

aA+b: =\{ax+b|x\in A\}

. If

a\neq0

then we calll

aA+b

and

A

to similar sets. In this question the Cantor set

C

is the number of non-negative integers that in their base-3 representation there is no

1

digit. You see

C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)

(i.e.

C

is partitioned to sets

3C

and

3C+2

). We give another example

C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)

. A representation of

C

is a partition of

C

to some similiar sets. i.e.

C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)

and

C_{i}=a_{i}C+b_{i}

are similar to

C

. We call a representation of

C

a primitive representation iff union of some of

C_{i}

is not a set similar and not equal to

C

. Consider a primitive representation of Cantor set. Prove that a)

a_{i}>1

. b)

a_{i}

are powers of 3. c)

a_{i}>b_{i}

d) (1) is the only primitive representation of

C

Problem Statement