Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A characterization theorem for the $L^{2}$-discrepancy of integer points in dilated polygons (1504.03251v1)

Published 2 Apr 2015 in math.NT

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.