I want to calculate the number of unique integer coordinate points inside an $n$-dimensional solid space. My approach is to approximate the number of points as the volume enclosed by the solid and then remove duplicates by considering the axes of symmetry.
eg. for $n=2$ let the equation of the curve be $(a+x) \times (a+y) \leqslant L$ where $x,y \geq 0$. So the number of within the area can be found by integration and we can remove the duplicates by considering the line of reflection passing thru the origin and cutting the hyperbola at the midpoint. Sample image:
So I want to generalize this solution for $n$-dimensions. For $n=3$ (when the equation is $(a+x)×(a+y)×(a+z) \leqslant L$). I am thinking of approximating the volume by considering a tetrahedron as integral is very hard to solve. But still the problem of removing duplicates is there.
One obvious axis symmetry cuts the tetrahedron in half but beyond that I don't know how I should proceed.
Thanks for any answers in advance.