Question

(Real Analysis)

Cantor set and Cantor ternary function Properties of Ck o C is closed Proposition 19 C is closed, uncountable, m(C) 0 p-nary expansion Let r E (0,1) and p a natural number with p as 1. Then r can be written where a e (0,1,2.. ,p-1) r- p" Proof for p 3 case: HW 36 Cantor set and Cantor ternary function Unique expression when p 3 x E (0, 1), p-3 x- , an E {0, 1,2) n-l 23 2 2 27 33 2,1,2)3 -7+ 2+ 23+ 4.A (2,1, 1,2, )a Fact. Only triadic numbers of the form All others have the expression a-X has exactly two expressions uniquely

Please prove for p=3 case with details.

Answer 1

p-nary expansion Let a e (0,1) and p a natural number with p1. Then a can be written as dn, where an E 0,1,2,. .p-1) 72=1 Proof for p 3 case: HW al talse a2 - 2

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