The distribution of lattice points on the four-dimensional sphere

Let $r_l(n)$ be the number of representations of $n$ by a sum of $l$ squares of integers and let $0<A<1$ be a constant. It is proved that if $(n,2)=1$, then $\sum_{-A\le w/\sqrt n\le A} r_3(n-w^2)=\mu_4(A)r_4(n)+O(n^{1487/2000}),\mu_4(A)>0$. Previously, the author obtained this asymptotics with a weaker error term $O(n^{3/4+\varepsilon})$.