If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of points inside an annulus of inner radius $R$ and area $1$ centered at the origin is $O(R^{\epsilon})$.

Is this bound still true if the annulus is not centered at the origin?

This seems harder to prove since when we are off the origin there isn't "much number theory" to help.

I am aware of averaged bounds for points inside an annuli off the origin, but I have not yet seen anything "pointwise". I would appreciate any known approaches or results on how to deal with that.

Thanks in advance

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of lattice points inside an annulus of inner radius $R$ and area $1$ centered at the origin is $O(R^{\epsilon})$.

Is this bound still true if the annulus is not centered at the origin?

This seems harder to prove since when we are off the origin there isn't "much number theory" to help.

I am aware of averaged bounds for points inside an annuli off the origin, but I have not yet seen anything "pointwise". I would appreciate any known approaches or results on how to deal with that.

Thanks in advance