This preprint was accepted February 12, 1998
Contact:
M.M.Skriganov and A.N.Starkov
ABSTRACT:
In the present paper we give an improvement of a previous result
of the paper [Skr, Theorem 2.2] on logarithmically small errors
in the lattice point problem for polyhedra. This improvement is
based on an analysis of hidden symmetries of the problem
generated by the Weyl group for $SL(n,\BR)$. Let $UP$ denote a
rotation of a given compact polyhedron $P\subset\BR^n$
by an orthogonal matrix $v\in SO(n)$, $tUP$ a dilatation of $UP$
by a parameter $t>0$, and $N(tUP)$ the number of integer points
$\ga\in\Bbb Z^n$ which fall into the polyhedron $tUP$. We show
that for almost all rotations $U$ (in the sense of the Haar
measure on $SO(n)$) the following asymptotic formula
$$
N(tUP)=t^n\text{\rm vol}\,P+ O((\log t)^{n-1+\eps}),\quad
t\to\infty,
$$
holds with arbitrary small $\eps>0$.