Undergrad Finite many Lattice Points in Sphere?

[Peter_Newman]

Hello,

I am wondering if in an n-ball the number of lattice points is finite.

First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big) ball that is smaller than the successive minimum. If we assume that there are infinitely many lattice points in the ball, wouldn't that amount to a contradiction, because the ball itself has a finite volume?

Is it even possible to argue like this? Or what would be an argument that the number of lattice points in the ball is finite?

 

[BvU]

What is a 'successive minimum' ?
What is an 'n-ball' ? Or do you mean an n-dimensional sphere ?

$\$

 

[Peter_Newman]

Hi @BvU, n-ball a.k.a n-dim sphere, right! Regarding the successive minimum, the first minimum is relevant, this is the length of the shortest vector, namely $\lambda_1$.

 

[BvU]

Never heard of the guy. Where does this $\lambda_1$ live ?
And once he/she is revealed, what is the second (successive ?) minimum ?

 

[hutchphd]

If we assume that there are infinitely many lattice points in the ball

How can you assume this??

 

[Peter_Newman]

How can you assume this??

My idea was to come to a contradiction by assuming that. Recap, I consider lattices from a number theory perspective...

 

[BvU]

If the number of lattice points increases, their inter-lattice-point distance decreases, and so does the volume of each of your little spheres. The product of number of spheres times volume never exceeds the total volume.

$\$

 

[Peter_Newman]

And since the product of number of spheres times volume never exceeds the total volume, we can say that there are only finite many lattice points in the n-dim. ball.