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Random fluctuations of convex domains and lattice points

Authors:	Alex Iosevich Kimberly K J Kinateder

Institution:	Department of Mathematics, Georgetown University, Washington, DC 20057 ; Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435

Abstract:	In this paper, we examine a random version of the lattice point problem. Let $\mathcal H$ denote the class of all homogeneous functions in $C^2(\mathbb R^n)$ of degree one, positive away from the origin. Let $\Phi$ be a random element of $\mathcal H$ , defined on probability space $(\Omega,\mathcal F,P)$ , and define $\begin{displaymath}F_{\Phi(\omega,\cdot)}(\xi)=\int _{\{x\colon\Phi(\omega,x)\le 1\}}e^{-i\langle x,\xi\rangle}dx\end{displaymath}$ for $\omega\in\Omega$ . We prove that, if $E(\|F_\Phi(\xi)\|)\le C\xi]^{\frac{n+1}{2}}$ , where $\xi]=1+\|\xi\|$ , then $\begin{displaymath}E(N_\Phi)(t)=Vt^n+O(t^{n-2+\frac{2}{n+1}})\end{displaymath}$ where $V=E(\|\{x\colon\Phi(\cdot,x)\le 1\}\|)$ , the expected volume. That is, on average, $N_\Phi(t)=Vt^n+O(t^{n-2+\frac{2}{n+1}})$ . We give explicit examples in which the Gaussian curvature of $\{x\colon \Phi(\omega,x)\le 1\}$ is small with high probability, and the estimate $N_\Phi(t)=Vt^n+O(t^{n-2+\frac{2}{n+1}})$ holds nevertheless.

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