Uniform distribution of integral points on multidimensional ellipsoids

Abstract

Let Q(X), XT=(x1,...,xl), be a positive definite, integral-valued, primitive, quadratic form of l⩾4 variables, let τ(η) be the number of solutions of Eq. Q(X)=n, let τ(η,Ω) be the number of the solution of the equation Q(X)=n such that X/√ɛΩ, where Ω is an arbitrary convex domain on Q(X)=1 with a piecewise smooth boundary. One investigates the asymptotic behavior of the quantity τ(η,Ω) (n→∞). In the case of an even l⩾4 the result is formulated in the following manner: for (n,N)=1 and n→∞ one has , e>o, whereμ(Ω) is the measure of the domain Ω, normalized by the conditionμ(E)=1, where E is the ellipsoid Q(X)=1. Weaker results have been obtained earlier by various authors. In the case of the simplest domains (“belt,” “cap”) the remainder in (1) can be brought to the form . The last estimate for large l is close to an unimprovable one. The proof makes use of the theory of modular forms and of Deligne's estimates.