Can I calculate an upper bound and a lower bound (or max or min) on the number of points that are uniformly distributed on a surface, knowing the area of the surface ?
More precisely, I have a sector of a disk in the shape of a pizza slice. I tile its surface using identical size hexagons. Because of the properties of the tiling, I know that the distance between the centers of any two adjacent hexagons is the same distance r where r is known. So, I guess I can say that the centers of hexagons are uniformly distributed (or equidistributed) on the surface of the pizza slice.
In this case can I calculate the maximum and minimum number of the centers of the hexagons that can be found on the surface of the pizza slice knowing the area of the pizza slice? or at least can I find some upper and lower bounds on the number of centers of hexagons?
Have a look at the following question and its answer, from which I've restated the answer.
gauss circle problem on a hexagonal lattice
Basically the answer is known for any regular crystallographic group, it seems. For your case, the number for a circle of radius $r$ and sector angle $\theta$ would seem to be $$ N(r;\theta,x,x_0)= \frac{\theta r^2}{\sqrt{3}} + O(r^{2/3} ), $$ as $r\to+\infty.$
Remark: This assumes a lattice point is at the origin, at which the circle out of which we're cutting a sector of angle $\theta$ is also centred.