Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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![Let F consist of those real numbers in [0,1] whose non-terminating
decimal expansions contain only the digits 0, 5 and 8. Thus if
Eo = [0, 1], then F = 0 Ek, where Ek+1 is constructed by removing
all but an appropriate choice of three tenths of each interval in Ek.
(a) Sketch, or give a detailed description of, the sets E1 and E2.
(b) Use parts (iii) and (v) of Equivalent definitions 2.1 in Falconer to
determine the box dimension of F.
(c) Use a Cantor set result from Chapter 4 of Falconer to show that
dim F = dim F.
(d) Let f [0,2] R be the function defined by
f(x)=(x+k), for k Є N.
Use the Mean Value Theorem to show that f is bi-Lipschitz, for
each kЄN, and hence determine
dimB Ufk(F).
k=1](https://content.bartleby.com/qna-images/question/a6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4/06087772-86b4-411c-b5c9-808210e08ac5/mvyp0a_thumbnail.png)
Transcribed Image Text:Let F consist of those real numbers in [0,1] whose non-terminating
decimal expansions contain only the digits 0, 5 and 8. Thus if
Eo = [0, 1], then F = 0 Ek, where Ek+1 is constructed by removing
all but an appropriate choice of three tenths of each interval in Ek.
(a) Sketch, or give a detailed description of, the sets E1 and E2.
(b) Use parts (iii) and (v) of Equivalent definitions 2.1 in Falconer to
determine the box dimension of F.
(c) Use a Cantor set result from Chapter 4 of Falconer to show that
dim F = dim F.
(d) Let f [0,2] R be the function defined by
f(x)=(x+k), for k Є N.
Use the Mean Value Theorem to show that f is bi-Lipschitz, for
each kЄN, and hence determine
dimB Ufk(F).
k=1
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