Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric and o-minimal Littlewood-Offord problems   The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero
vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly
random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have
$\Pr(a_1\xi_1+\dots+a_n\xi_n=x)=O(n^{-1/2})$. In this paper we show that
$\Pr(a_1\xi_1+\dots+a_n\xi_n\in S)\le n^{-1/2+o(1)}$ whenever $S$ is definable
with respect to an o-minimal structure (for example, this holds when $S$ is any
algebraic hypersurface), under the necessary condition that it does not contain
a line segment. We also obtain an inverse theorem in this setting.

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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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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: physics