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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Random fluctuations of convex domains
and lattice points


Authors: Alex Iosevich and Kimberly K. J. Kinateder
Journal: Proc. Amer. Math. Soc. 127 (1999), 2981-2985
MSC (1991): Primary 42Bxx; Secondary 60G99
Published electronically: April 27, 1999
MathSciNet review: 1605972
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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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Additional Information

Alex Iosevich
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057
Email: iosevich@math.georgetown.edu

Kimberly K. J. Kinateder
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
Email: kjk@euler.math.wright.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04879-0
PII: S 0002-9939(99)04879-0
Received by editor(s): September 17, 1997
Received by editor(s) in revised form: January 6, 1998
Published electronically: April 27, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society