Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the number of ways of writing a natural number $n$ as the sum of two squares. My question is then, what happens if the circle is not centered at the origin (or any lattice point)? In this case it seems that there is no number theoretical exact formula to reduce one problem to the other, and we have to tackle the GCP directly. What are the recent results in this direction?

**Addendum**: I vaguely recall reading somewhere about the same problem with the circle replaced by a uniformly convex planar domain, in which case the exponent is not as good, but still better than $O(R^{2/3})$. But now I'm unable to find any reference to it, though, so I added a "reference request" tag.