Random fluctuations of convex domains and lattice points

In this paper, we examine a random version of the lattice point problem. Le t H denote the class of all homogeneous functions in C-2 (R-n) of degree on e, positive away from the origin. Let Phi be a random element of H, defined on probability space (Omega; F; P), and define F-Phi(omega; .) (xi) = integral({x: Phi(omega,x) less than or equal to 1}) e (-i [x, xi]) dx for omega is an element of Omega. We prove that, if E(\F-Phi(xi)\) less tha n or equal to C[xi] n+1/2, where [xi] = 1 + \xi\, then E(N-Phi)(t) = Vt(n) +O(t(n-2+) 2/n+1) where V = E(\{x: Phi(.; x) less than or equal to 1}\), the expected volume. That is, on average, N-Phi (t) = Vt(n) +O(t(n-2+) 2/n+1) We give explicit examples in which the Gaussian curvature of {x: Phi(omega, x) less than or equal to 1} is small with high probability, and the estimate N-Phi (t) = Vt (n) +O(t(n-2+) 2/n+1) holds nevertheless.