Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets
  in $\mathbb{R}^n$   In his 1990 paper, Jones proved the following: given $E \subseteq
\mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and
\[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E + \sum_{Q}
\beta_{E}(3Q)^2\ell(Q).\] Here, $\beta_E(Q)$ measures how far $E$ deviates from
a straight line inside $Q$. This was extended by Okikiolu to subsets of
$\mathbb{R}^n$ and by Schul to subsets of a Hilbert space.
  In 2018, Azzam and Schul introduced a variant of the Jones $\beta$-number.
With this, they, and separately Villa, proved similar results for lower regular
subsets of $\mathbb{R}^n.$ In particular, Villa proved that, given $E \subseteq
\mathbb{R}^n$ which is lower content regular, there exists a `nice'
$d$-dimensional surface $F$ such that $E \subseteq F$ and \begin{align}
\mathscr{H}^d(F) \sim \text{diam}( E)^d + \sum_{Q} \beta_{E}(3Q)^2\ell(Q)^d.
\end{align} In this context, a set $F$ is `nice' if it satisfies a certain
topological non degeneracy condition, first introduced in a 2004 paper of
David.
  In this paper we drop the lower regularity condition and prove an analogous
result for general $d$-dimensional subsets of $\mathbb{R}^n.$ To do this, we
introduce a new $d$-dimensional variant of the Jones $\beta$-number that is
defined for any set in $\mathbb{R}^n.$

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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg