# What does it mean by saying that a number is 'asymptotic to ' another number?

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.

• You should consider $V$ as a varying value, and consider how the number of points evolves as $V$ goes to infinity. – zozoens Jan 27 '15 at 10:30
• @bof : You are right! Thx for pointing that out. – 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 $1$.
• 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! – Nighty Jan 27 '15 at 11:40
• 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 $\Delta (K)$. – quid Jan 27 '15 at 11:51