We show that if s is any integer with s f ð10RÞ g , where g ¼ ð10RÞ 5 , then given any R real cubic forms C 1 ; . . . ; C R in s variables, there is a nonzero integral solution x of the simultaneous Diophantine inequalities jC 1 ðxÞj < 1; jC 2 ðxÞj < 1; . . . ; jC R ðxÞj < 1.

Abstract. We treat systems of real diagonal forms F 1 (x), F 2 (x), . . . , F R (x) of degree k, in s variables. We give a lower bound s 0 (R, k), which depends only on R and k, such that if s ≥ s 0 (R, k) holds, then, under certain conditions on the forms, and for any positive real number , there is a nonzero integral simultaneous solution x ∈ Z s of the system of Diophantine inequalities |F i (x)| < for 1 ≤ i ≤ R. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.