Any bounded planar region with positive area 
placed in any position of the unit square lattice can be
translated so that the number of lattice
points inside the region will be at least 
(Blichfeldt 1914, Steinhaus 1999). The theorem can be generalized
to 
dimensions.
Blichfeldt's Theorem
See also
Lattice Point, Minkowski Convex Body Theorem, Pick's TheoremExplore with Wolfram|Alpha
References
Blichfeldt, H. F. "A New Principle in the Geometry of Numbers, with Some Applications." Trans. Amer. Math. Soc. 15, 227-235, 1914.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 97-99, 1999.Referenced on Wolfram|Alpha
Blichfeldt's TheoremCite this as:
Weisstein, Eric W. "Blichfeldt's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BlichfeldtsTheorem.html