The following comes from the book The Geometry of Numbers by C.D. Olds, Anneli Lax and Giuliana Davidoff. After the discussion of the geometry of numbers, its application, lattice-point packing is mentioned.

Let $K$ be a convex set, or body, symmetrically placed about the origin O.


Now let us consider the density of such a packing in an arbitrary admissible lattice(1), that is, the proportion of space occupyed by the translates of $\frac12 K$. Denote by $V(\frac12 K)$ the volume of each of the translates. In a large cube of volume $V$, the number of lattice points is asymptotic to $V / \Delta$, where $\Delta$ is the volume of the fundamental domain of the admissible lattice under consideration.

(1) : An admissible lattice for $K$ is a lattice that has no lattice point inside $K$ other than origin.

I am confused with the term 'asymptotic to'. I first came across this term when I learnt curve-sketching in calculus. Let's take a look at a cube in the usual $\mathbb{R}^3$. If I am correct, $\Delta=1$. If I consider a $3 \times 3 \times 3$ cube with a vertex placed at origin, then $V=27$ and the number of lattice points within the cube or on the boundary would be $64$. Now, what asymptotic property are we looking at?

Any help would be appreciated.

  • $\begingroup$ You should consider $V$ as a varying value, and consider how the number of points evolves as $V$ goes to infinity. $\endgroup$ – zozoens Jan 27 '15 at 10:30
  • $\begingroup$ @bof : You are right! Thx for pointing that out. $\endgroup$ – Nighty Jan 27 '15 at 12:04

One says that two quantities/functions $f(t)$ and $g(t)$, depending on a prameter $t$, are asymptotically equal (as $t \to \infty$) when the limit of $f(t)/g(t)$ is $1$.

Considering larger and larger cubes you will see that the quotient of the two quantities you consider will tend to one. (The absolute difference might grow however; this is not a contradiction.)

  • $\begingroup$ Thanks for replying! Could you also explain this to me? 'If $K$ is a set with at least one admissible lattice, we take $\Delta (K)=\inf \Delta (L)$ where the infimum is taken over all admissible lattices for $K$.' I don't understand what it is talking about! $\endgroup$ – Nighty Jan 27 '15 at 11:40
  • $\begingroup$ A lattice is admissible for $K$ if there is not lattice point (other than $0$) in $K$. To give an example: if you consider the lattice of all integer points, then this lattice is not admissible for $K$ a disk of radius $3/2$ since this contains $(1,0)$ for instance. Yet if you consider the lattice of all even integers than it is admissible as the disc does not contain any point with even integral coordinates, other than $(0,0)$. Among all lattices that are admissible for some $K$ you try to find which is the smalles possible $\Delta$ of the lattice and this is the $\Delta (K)$. $\endgroup$ – quid Jan 27 '15 at 11:51
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    $\begingroup$ That's very detail! Thx a lot! $\endgroup$ – Nighty Jan 27 '15 at 12:04

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