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.
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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.
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.)