16
$\begingroup$

Suppose we have a sphere centered at the origin of $\mathbb{R^{n}}$ with radius $r$. Are there known theorems that state the number of integer lattice points that lie on the sphere? It seems like this is something someone has studied so hopefully someone here could point me to some references.

Also, consider the lattice points that do lie on this sphere. Is there a known greatest lower bound to the number of neighbours (integer lattice points lying Euclidean distance $1$ away) of these lattice points that do not lie in the sphere as a function of $r$ and $n$?

  • 1
    The 2D problem is known as the [Gauss Circle Problem](http://en.wikipedia.org/wiki/Gauss%27s_circle_problem). I think even less is known about the 3D version.2011-11-27
  • 3
    I think the Gauss Circle Problem pertains to the number of points that lie inside the sphere, just not the ones on the surface.2011-11-27
  • 0
    You're right. Not the same thing (but related). :)2011-11-27
  • 0
    There are effective estimates known derived using Harmonic analytic techniques, see [here](http://www.uam.es/personal_pdi/ciencias/fchamizo/kiosco/files/lpha.pdf).2011-11-27
  • 0
    @RagibZaman: Maybe I skimmed it too fast, but it seems like this is also talking about lattice points inside a shape, not just on the surface. Sorry, if I am wrong.2011-11-27
  • 1
    See http://mathworld.wolfram.com/SumofSquaresFunction.html2011-11-28

2 Answers 2