Fourth Homework Set Answers

10. (a) The gasket has dimension Log(3)/Log(2) and the Cantor middle thirds set has dimension Log(2)/Log(3). By the product formula,

d(G × C) = d(G) + d(C) = Log(3)/Log(2) + Log(2)/Log(3)

This is about 1.585 + 0.631, greater than 2.

(b) The Cantor set C(r) has dimension Log(2)/Log(1/r). Applying the product formula, we want

Log(3)/Log(2) + Log(2)/Log(1/r) = 2.

That is,

Log(2)/Log(1/r) = 2 - Log(3)/Log(2) = 2Log(2)/Log(2) - Log(3)/Log(2) = Log(4)/Log(2) - Log(3)/Log(2) = Log(4/3)/Log(2)

Cross-multiplying gives

(Log(2))2 = Log(4/3)Log(1/r)

That is,

Log(1/r) = (Log(2))2/Log(4/3)

So

1/r = 10(Log(2))2/Log(4/3)

This gives

r = 1/(10(Log(2))2/Log(4/3))

This is about 0.188.

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