Revision of Use self-similarity to get a limit from an inferior or superior limit. from Thu, 26/08/2010 - 13:51

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In some situations the existence of a limit can be derived by using inferior or superior limit and suitably dividing a domain to use self-similarity.

Prerequisites

Basic real analysis.

Example 1: Maximal density of a packing

Fix a compact domain $D$ in Euclidean space $\mathbb{R}^n$ (for example, a ball). A packing $P$ is then a union of domains congruents to $D$ , with disjoint interiors. The density of a packing is defined as

$\delta(P)=\lim_{r\to \infty} \frac{V(P\cap B(r))}{V(B(r))}$

where $V$ is the volume and $B(r)$ is the square $[-r,r]^n$ , when the limit exists.

The present trick shows that if $P_r$ is a packing contained in $B(r)$ of maximal volume, then $\lim_{r\to \infty} \delta(P_r)$ exists.

First, since $\delta(P_r)\in [0,1]$ , the superior limit $\ell=\limsup_{r\to \infty} \delta(P_r)$ exists. We want to prove that the inferior limit equals the superior one. Given any $\varepsilon>0$ , there is an $r_0$ such that $\delta(P_{r_0})>\ell-\varepsilon$ . Now for any integer $k$ the square $B(k r_0)$ can be divided into $k^n$ translates of $B(r_0)$ . Each of these contains a packing of density greater than $\ell-\varepsilon$ , so that $\delta(P_{kr_0})>\ell-\varepsilon$ for all $k$ . This rewrites as $V(P_{k r_0}) > (2kr_0)^n(\ell-\varepsilon)$ . By maximality, $V(P_r)$ is non-decreasing in $r$ ; for all $r>r_0$ one can introduce $k=\lfloor r/r_0\rfloor$ and gets $V(P_r)>(2kr_0)^n(\ell-\varepsilon)$ . It follows that

$\delta(P_r)>\left(1-\frac{r_0}{r}\right)^n (\ell-\varepsilon)$

so that as soon as $r$ is large enough, $\delta(P_r)>\ell-2\varepsilon$ . As a consequence, $\liminf_r \delta(P_r)\geq \limsup_r \delta(P_r)$ and we are done.

Example 2: Weyl's inequality and polynomial equidistribution

This example is taken from a mathoverflow question and answer.