Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $k$-point configuration sets with nonempty interior   We give conditions for $k$-point configuration sets of thin sets to have
nonempty interior, applicable to a wide variety of configurations. This is a
continuation of our earlier work \cite{GIT19} on 2-point configurations,
extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in
Euclidean spaces. We show that for a general class of $k$-point configurations,
the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has
nonempty interior provided that the sum of their Hausdorff dimensions satisfies
a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated
generalized Radon transforms over all nontrivial partitions of the $k$ points
into two subsets. We illustrate the general theorems with numerous specific
examples. Applications to 3-point configurations include areas of triangles in
$\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned
parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb
R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios
on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb
R^2$, and dot products of differences in $\mathbb R^d$.

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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci