Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\epsilon}$, $\forall \epsilon>0$.

Now I have a question about this result. We consider a $k$-sphere $S^k(a)$ embedded in $\mathbb{R}^d$ with arbitrary center $a\in\mathbb{R}^d$ and radius $R$, where $d\ge k+1$. Do we have the same estimates about the size of $\mathbb{Z}^{d}\cap S^k(a)$? Can we have uniform estimates independent of the center $a$ and dimension $d$? I don't know if there are any references about this generalization. Any help and comments will be greatly appreciated.