Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How likely can a point be in different Cantor sets   Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_\lambda: \lambda\in(0,
1/m]\}$ be a class of Cantor sets, where $K_{\lambda}=\{\sum_{i=1}^\infty
d_i\lambda^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this
paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More
precisely, for a fixed point $x\in(0,1)$ we consider the parameter set
$\Lambda(x)=\{\lambda\in(0,1/m]: x\in K_\lambda\}$, and show that $\Lambda(x)$
is a topological Cantor set having zero Lebesgue measure and full Hausdorff
dimension. Furthermore, by constructing a sequence of Cantor subsets with large
thickness in $\Lambda(x)$ we prove that the intersection
$\Lambda(x)\cap\Lambda(y)$ also has full Hausdorff dimension for any $x,
y\in(0,1)$.

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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nlin