I have a circle of radius $r$, and I wish to place this circle of a $Z^2$ integer lattice or an $A_2$ hexagonal lattice s.t. I maximize the number of lattice points within or along the contour of the circle.

Trivially, for $r < \frac{R}{2}$, where $R$ is the smallest spacing between lattice points, it will be optimal to center the circle on a lattice point. However, is there an optimal position within the fundamental unit of either lattice if $r \geq T$, where $T$ is some threshold value?

Taking a guess for the $Z^2$ integer lattice, I'd say that centering the circle on a lattice point will be optimal for $r \geq \sqrt(2)$ and $r < \frac{1}{2}$. For $\frac{\sqrt(2)}{2} \leq r < \sqrt(2)$, the optimal position for the circle will be in the center of the square unit cell of the lattice (allowing coverage of four lattice points), and for $\frac{1}{2} \leq r < \frac{\sqrt(2)}{2}$, the optimal position will between two lattice points separated by the smallest possible inter-point distance (i.e. the unit distance) s.t. the circle covers these two points. However, I don't know how to prove this.

For an $A_2$ hexagonal lattice, we know for $r < \frac{1}{2}$ that centering the circle on a lattice point will be optimal (allowing coverage of a single lattice point), and can guess that the next optimal position will be between two lattice points separated by the smallest possible inter-point distance (i.e. the unit distance) for $\frac{1}{2} \leq r < \frac{1}{\sqrt{3}}$ (covering two lattice points), then at the barycenter of an equilateral triangle cell in the lattice for $\frac{1}{\sqrt{3}} \leq r < \frac{\sqrt{3}}{2}$ (covering three lattice points), then, it seems, back between two lattice points separated by the unit distance for $\frac{\sqrt{3}}{2} \leq r < 1$ (covering four lattice points), then back to the center of the lattice for $r = 1$ (covering seven lattice points)? Is it optimal to remain centered on a lattice point for $r > 1$?

In a previous question ( An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice ) I noted the exact counting solution for the number of points contained or along the contour of a circle of radius $r$ centered on a lattice point in a $Z^2$ integer lattice, and Yoav Kallus provided an exact counting solution for the $A_2$ hexagonal lattice.

We can note that, if $D$ is the largest spacing between lattice points in an arbitrary lattice ($D = \sqrt(2)$ on a $Z^2$ integer lattice and $D = 1$ on an $A_2$ hexagonal lattice), then a circle of radius $(r + D)$ centered on a lattice point will always cover more lattice points than a circle of radius $r$ centered at arbitrary coordinates.