Type: Article
Publication Date: 1993-07-01
Citations: 237
DOI: https://doi.org/10.1215/s0012-7094-93-07107-4
A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V)= {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of A-points of V. Hence I1" is some Euclidean norm on R".The only general method available for such problems is the Hardy-Littlewood circle method, which however has certain limitations, requiring roughly that the codimension of V in the ambient space A", as well as the degree of the equations (1.1), be small relative to n.Furthermore, there are restrictions on the size of the singular sets of the related varieties:V u {x e C": f(x) &, j 1,..., v}, u () e c".We refer to [Bi] and [Sch] for a discussion of the restriction.Regardless of these restrictions, one hopes that for many more cases N(T, V) can be given in the form predicted by the Hardy-Littlewood method, that is, as a product oflocal densities: (,) N(T, V) l--I l,(V)lUoo( T, v), p <oo where the "singular series" I-I,< #,(V) is given by p-adic densities:and/(T, V) is a real densitymthe "singular integral."Following Schmidt [Sch], we say that V is a Hardy-Littlewood system if the above asymptotics (,) is valid.