Finite many Lattice Points in Sphere?

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Discussion Overview

The discussion revolves around the question of whether the number of lattice points within an n-dimensional ball (n-ball) is finite. Participants explore theoretical arguments and mathematical reasoning related to the properties of lattice points and their distribution within a bounded volume.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that if there are infinitely many lattice points in a bounded n-ball, it would lead to a contradiction due to the finite volume of the ball.
  • Another participant questions the definition of 'successive minimum' and the concept of an n-ball, seeking clarification on these terms.
  • A participant clarifies that the first successive minimum, denoted as ##\lambda_1##, is the length of the shortest vector in the lattice.
  • Further inquiries are made about the existence and definition of the second successive minimum.
  • One participant challenges the assumption of infinitely many lattice points, prompting a discussion on the implications of this assumption.
  • Another participant argues that if the number of lattice points increases, the distance between them decreases, which would affect the volume of associated spheres, suggesting a limit on the number of lattice points.
  • A later reply concludes that since the product of the number of spheres and their volume cannot exceed the total volume of the n-ball, there must be only finitely many lattice points.

Areas of Agreement / Disagreement

Participants express differing views on the assumption of infinitely many lattice points, with some questioning its validity while others attempt to derive conclusions based on that assumption. The discussion remains unresolved regarding the finiteness of lattice points in the n-ball.

Contextual Notes

Participants reference mathematical concepts such as lattice points and successive minima without fully establishing definitions or resolving the implications of their assumptions. The discussion involves complex reasoning that may depend on specific mathematical frameworks.

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

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

##\ ##
 
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.
 

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