Exact Hausdorff Measure of Certain Non-Self-Similar Cantor Sets
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
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arXiv subjects
Jason D. Phillips
Publications and source records attributed to Jason D. Phillips.
On Intersections of Cantor Sets: Self-Similarity
Let C be a Cantor set. For a real number t let C+t be the translate of C by t, We say two real numbers s,t are equivalent if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets determined by similarities with one fixed positive contraction ratio. For this class of Cantor set, we show that an "initial segment" of the intersection of C and C+t is a self-similar set with contraction ratios that are powers of the contraction ratio used to describe C as a self- similar set if and only if t is equivalent to a rational number. Our results are new even for the middle thirds Cantor set.
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On Intersections of Cantor Sets: Hausdorff Measure
We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.
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