Does there exist a branch of mathematics that specifically study the number of lattices enclosed by a region? I have seen that sometimes, in particular in number theory and combinatorial commutative algebra, our questions are somehow related to finding the number of points with integer components in a region/shape. Since this seems to be related to some open problems and conjectures, does there exist a branch of mathematics that specifically study such questions? If yes, what it's called and what prerequisites one must know to study it.
 A: For a square lattice, if you're counting the number of lattice points inside a polygon whose vertices are lattice points, then you can use Pick's theorem.  Other relevant theorems are Blichfeldt's theorem and the Minkowski Convex Body theorem.
Some Wikipedia links that might help:
Lattice (group),
Integer points in convex polyhedra
A: In case that some people are interested to know more about 'geometry of numbers' I found some good books on the subject. Some of them were too long and advanced, so they require that you devote a good time on this subject, but the best book I found on this topic is "The geometry of numbers" written by C.D. Olds, Anneli Lax and Giuliana P. Davidoff which has been published by The Mathematical Association of America.
The topics that it covers includes:
Part I - Lattice points and number theory
Chapter 1 - Lattice points and Straight Lines
Chapter 2 - Counting Lattice points
Chapter 3 - Lattice points and the area of Polygons
Chapter 4 - Lattice points in Circles
Part II - An introduction to the geometry of numbers
Chapter 5 - Mikowski's fundamental theorem
Chapter 6 - Applications of Minkowski's theorems
Chapter 7 - Linear transformations and Integral Lattices
Chapter 8 - Geometric interpretations of Quadratic forms
Chapter 9 - A new principle in the Geometry of numbers
Chapter 10- A Minkowski Theorem(Optional)
Appendix I   - Gaussian Integers. by Peter D. Lax
Appendix II  - The closest packing of Convex Bodies
Appendix III - Brief biographies
