Random fluctuations of convex domains and lattice points

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 \[ F_{\Phi (\omega ,\cdot )}(\xi )=\int _{\{x\colon \Phi (\omega ,x)\le 1\}}e^{-i\langle x,\xi \rangle }dx\] for $\omega \in \Omega$. We prove that, if $E(|F_\Phi (\xi )|)\le C[\xi ]^{\frac {n+1}{2}}$, where $[\xi ]=1+|\xi |$, then \[ E(N_\Phi )(t)=Vt^n+O(t^{n-2+\frac {2}{n+1}})\] 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.