Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characterization theorem for the $L^{2}$-discrepancy of integer points
  in dilated polygons   Let $C$ be a convex $d$-dimensional body. If $\rho$ is a large positive
number, then the dilated body $\rho C$ contains $\rho^{d}\left\vert
C\right\vert +\mathcal{O}\left( \rho^{d-1}\right) $ integer points, where
$\left\vert C\right\vert $ denotes the volume of $C$. The above error estimate
$\mathcal{O}\left( \rho^{d-1}\right) $ can be improved in several cases. We are
interested in the $L^{2}$-discrepancy $D_{C}(\rho)$ of a copy of $\rho C$
thrown at random in $\mathbb{R}^{d}$. More precisely, we consider \[
D_{C}(\rho):=\left\{ \int_{\mathbb{T}^{d}}\int_{SO(d)}\left\vert
\textrm{card}\left( \left( \rho\sigma(C)+t\right) \cap\mathbb{Z}^d\right) -
\rho^{d}\left\vert C\right\vert \right\vert ^{2}d\sigma dt\right\} ^{1/2}\ , \]
where $\mathbb{T}^{d}=$ $\mathbb{R}^{d}/\mathbb{Z}^{d}$ is the $d$-dimensional
flat torus and $SO\left( d\right) $ is the special orthogonal group of real
orthogonal matrices of determinant $1$.
  An argument of D. Kendall shows that $D_{C}(\rho)\leq c\ \rho^{(d-1)/2}$. If
$C$ also satisfies the reverse inequality $\ D_{C}(\rho)\geq c_{1} \
\rho^{(d-1)/2}$, we say that $C$ is $L^{2}$\emph{-regular}. L. Parnovski and A.
Sobolev proved that, if $d>1$, a $d$-dimensional unit ball is $L^{2}% $-regular
if and only if $d\not \equiv 1\ (\operatorname{mod}4)$.
  In this paper we characterize the $L^{2}$-regular convex polygons. More
precisely we prove that a convex polygon is not $L^{2}$-regular if and only if
it can be inscribed in a circle and it is symmetric about the centre.

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The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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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: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat