The Pointwise Densities of the Cantor Measure |
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Authors: | De-Jun Feng Su Hua Zhi-Ying Wen |
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Institution: | Department of Mathematical Science and Center for Advanced Study, Tsinghua University, Beijing, 100084, People's Republic of Chinaf1;Department of Mathematical Science, Tsinghua University, Beijing, 100084, People's Republic of China, f2;Department of Mathematical Science, Tsinghua University, Beijing, 100084, People's Republic of China, f3;d Department of Mathematics, Wuhan University, Wuhan, 430072, People's Republic of China |
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Abstract: | Let
be the classical middle-third Cantor set and let μ
be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x
the upper and lower s-densities Θ*s(μ
, x), Θ*s(μ
, x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F
such that 9(Θ*s(μ
, x))− 1/s + (Θ*s(μ
, x))− 1/s = 16 holds for x
\F. Furthermore, for μC almost all x, Θ*s(μ
, X) − 2 · 4− s and Θ*s(μ
, x) = 4− s. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set
is 4s. |
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Keywords: | Cantor measure upper and lower density packing measure |
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