On Simultaneous Diagonal Inequalities

Let F 1 , …, F t be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities | F 1 (x)|<τ,…,| F t (x)|<τ in integers x 1 , …, x s has been considered by a number of authors over the last quarter‐century, starting with the work of Cook [ 9 ] and Pitman [ 13 ] on the case t = 2. More recently, Brüdern and Cook [ 8 ] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F 1 , …, F t satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F 1 , …, F t to have different degrees. However, with the recent work of Wooley [ 18 , 20 ] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [ 18 ] and [ 20 ]. Let λ 1 , …, λ s and μ 1 , …, μ s be real numbers such that for each i either λ i or μ i is nonzero. We define the formsF ( x ) = λ 1 x 1 3 + … + λ s x s 3G ( x ) = μ 1 x 1 2 + … + λ s x s 2and consider the solubility of the system of inequalities| F ( x ) | <( max | x i | )‐ σ 1| G ( x ) | <( max | x i | )‐ σ 2      ( 1 )in rational integers x 1 , …, x s . Although the methods developed by Wooley [ 19 ] hold some promise for studying more general systems, we do not pursue this in the present paper. We devote most of our effort to proving the following theorem.