Finite many Lattice Points in Sphere?

In summary, the conversation discusses the possibility of an infinite number of lattice points within an n-ball and whether this is feasible given the properties of lattices and their inter-lattice-point distances. It is argued that an infinite number of lattice points would lead to a contradiction, therefore it can be concluded that the number of lattice points in an n-ball is finite.
  • #1
Peter_Newman
155
11
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? :angel:
 
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  • #2
What is a 'successive minimum' ?
What is an 'n-ball' ? Or do you mean an n-dimensional sphere ?

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  • #3
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##.
 
  • #4
Never heard of the guy. Where does this ##\lambda_1## live ?
And once he/she is revealed, what is the second (successive ?) minimum ?
 
  • #5
Peter_Newman said:
If we assume that there are infinitely many lattice points in the ball
How can you assume this??
 
  • #6
hutchphd said:
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...
 
  • #7
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.

##\ ##
 
  • #8
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.
 

1. What is the definition of "Finite many Lattice Points in Sphere"?

"Finite many Lattice Points in Sphere" refers to the number of points with integer coordinates that lie on the surface of a sphere with a given radius. These points are often referred to as "lattice points" because they can be thought of as the intersections of a lattice or grid on the surface of the sphere.

2. How is the number of lattice points in a sphere calculated?

The number of lattice points in a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume of the sphere and r is the radius. This formula is based on the fact that each lattice point represents a unit cube with volume 1, and the volume of the sphere is equal to the sum of all these unit cubes.

3. Can the number of lattice points in a sphere be infinite?

No, the number of lattice points in a sphere is always finite. This is because as the radius of the sphere increases, the volume also increases, but the number of lattice points remains constant. Therefore, as the radius approaches infinity, the density of lattice points on the surface of the sphere approaches zero.

4. What is the significance of studying lattice points in spheres?

Studying lattice points in spheres has applications in various fields such as mathematics, physics, and computer science. For example, in number theory, the distribution of lattice points in spheres can provide insights into the behavior of prime numbers. In physics, lattice points can represent atoms in a crystal structure. In computer science, lattice points can be used to generate random numbers for simulations and algorithms.

5. Are there any open questions or unsolved problems related to lattice points in spheres?

Yes, there are still many open questions and unsolved problems related to lattice points in spheres. One of the most famous is the Gauss circle problem, which asks for the exact number of lattice points that lie within a given circle. Other open questions include the distribution of lattice points in higher dimensions and the behavior of lattice points in non-Euclidean geometries.

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