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Media type:
E-Article
Title:
Random Fluctuations of Convex Domains and Lattice Points
Contributor:
Iosevich, Alex
imprint:
American Mathematical Society, 1999
Published in:Proceedings of the American Mathematical Society
Language:
English
ISSN:
0002-9939;
1088-6826
Origination:
Footnote:
Description:
<p>In this paper, we examine a random version of the lattice point problem. Let H denote the class of all homogeneous functions in C<sup>2</sup>( R<sup>n</sup>) of degree one, positive away from the origin. Let Φ be a random element of H, defined on probability space (Ω, F, P), and define F<tex-math>$_{\Phi (\omega,\cdot)}(\xi)=\int_{\{x \colon \Phi (\omega,x)\leq 1\}}e^{-i\langle x,\xi \rangle}dx$</tex-math>for ω ∈ Ω . We prove that, if E(|F<sub>Φ</sub>(ξ)|) ≤ C[ξ]<sup>n+1/2</sup>, where [ξ]=1+|ξ|, then E(N<sub>Φ</sub>)(t)=Vt<sup>n</sup>+O(t<tex-math>$^{n-2+\f rac{2}{n+1}}$</tex-math>) where V = E(|{x : Φ (·,x)≤ 1}|), the expected volume. That is, on average, N<sub>Φ</sub>(t) = Vt<sup>n</sup>+ O(t<sup>n-2+2/n+1</sup>). We give explicit examples in which the Gaussian curvature of {x: Φ (ω , x) ≤ 1} is small with high probability, and the estimate N<sub>Φ</sub>(t) = Vt<sup>n</sup>+ O(t<sup>n-2+2/n+1</sup>) holds nevertheless.</p>