The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"

For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\Omega(n^{k/2-1})$.

This approximation is very bad in limit with dimension. For example, if $n$ grows linearly with $k$ then the approximation term dominates the volume term. Do results exist for fixed $n$ and $k$ large?

The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"

For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\Omega(n^{k/2-1})$.

This approximation is very bad in limit with dimension. For example, if $n$ grows linearly with $k$ then the approximation term dominates the volume term. Do results exist for this regime, or at least for $n$ small relative to $k$?