Distribution of integral points on some hyperboloids of one sheet

Abstract

Results of B. F. Skubenko (Izv. Akad. Nauk SSSR, Ser. Mat.,26, 721–752 (1962)) are generalized to indefinite ternary quadratic formsf(x)=f 0(Cx), which are contained in the simplest formf 0(x)=x 1 x 3−x e r We prove that the integral points on the hyperboloid of one sheetf(x)=m,m<0,\(\sqrt {\left| m \right|} \notin Q\) are uniformly distributed over area (in the sense of hyperbolic metric) and over residue classes for given modulus.