Lattice point theory of irractional ellipsoids with an arbitrary center

Abstract

We shall consider positive definite quadratic formsQ inr≥2 variables of the “almost diagonal shape”$Q{\mathbf{ }}(u) = \sum\limits_{1 \leqslant j{\mathbf{ }} \leqslant \sigma } {\alpha j{\mathbf{ }}(u^{(j)} )}$ where ϖ≥2, and for 1≤j≤ϖ,Q _j is a positive definite quadratic form with integral coefficients inr _j variables, α _j is a positive real number,r _j≥1 andr ₁+...+r _α=r Letb ₁,...,b _r be a system of real numbers with 0≤b _j<1. For x>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)≤x, letV(x) be the volume of this ellipsoid and letP(x)=A(x)-V(x). Our purpose is to find the exact order ofP(x); i. e., the numberf for which for each ɛ>0P(x)=O(x^f+ε) andP(x)=Ω(x ^f−ε).