Algebra & Trigonometry with Analytic Geometry
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
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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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Algebra & Trigonometry with Analytic Geometry
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ISBN:9781133382119
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Publisher:Cengage