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 $k$? 



The Gauss Circle Problem asymptotic in dimension
This problem was studied by J.E. Mazo and A.M. Odlyzko in [Lattice Points in High-Dimensional Spheres](http://www.dtc.umn.edu/~odlyzko/doc/arch/high.dim.spheres.pdf). If $n$ grows with $k$ faster than linearly then the volume asymptotic still applies. If it increases slower than linearly then for most locations of the center of the circle there will be no lattice points inside it. The linear increase mentioned in the OP is a borderline case. For $n=\alpha k$ the asymptotics of the number of points $N$ inside the circle is 
$$\lim\_{k\rightarrow\infty}N=e^{ck},$$
with $c$ a coefficient of order unity that depends on $\alpha$ and on the location of the center of the circle. (For a circle centered at the origin and $\alpha=1$ the value is $c=1.418938538$.)


