Let F1, …, Ft
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
formula here
in integers x1, …, xs
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 F1, …, Ft
satisfy certain additional conditions. What has not yet been considered is the
possibility of allowing the forms F1, …, Ft
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
λ1, …, λs and
μ1, …, μs be real numbers such that for each
i either λi or μi is nonzero.
We define the forms
formula here
and consider the solubility of the system of inequalities
formula here
in rational integers x1, …, xs.
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.