  In this paper we mainly study sums of four rational squares with certain
restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational
numbers. We establish the following four-square theorem for rational numbers:
For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be
written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that
$ax+by+cz+dw$ is a rational square (or a rational cube). This paper also
contains many conjectures; for example, for any positive integers $a$ and $b$
with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be
written as $aw^4+bx^4+y^2+z^2$ with $w,x,y,z\in\mathbb Q$.
