When the origin is $(0,0)$, the number of lattice points on the circumference is directly related to the sum of squares function. And sum of square function can be arbitrary large according to its formula. I am wondering if the same holds for a more general case: given a rational origin, then either for arbitrary large $n$, there exist a circle with radius $r$ such that the number of lattice points on its circumference is larger than $n$; or there can only be at most $n$ points regardless or $r$.
My guess is the former, but my attempts on proving it with similar methods as $(0,0)$ case failed. Any help or reference is appreciated.