Exercise 23, § 4

On the space $\mathbf{R}^n$, let $\mu$ be Lebesgue measure, $|x|$ a norm such that the unit ball for this norm has measure equal to 1, and $(x_k)_{k \geqslant 1}$ an infinite sequence of distinct points of a bounded integrable set B such that $\mu(B) = 1$. For every integer m, denote by $d_m$ the infimum of the numbers $|x_i - x_j|$ for $1 \leqslant i < j \leqslant m$. Show that

where

(Argue by contradiction, assuming that for some $\varepsilon > 0$ there exists an $m_0$ such that $md_m^n > h_n$ for $m \geq m_0$, with $h_n = \alpha_n^{-1} + \varepsilon$. For $1 \leq i < m$, let $B_i$ be the ball with center $x_i$ and radius $\frac{1}{2} h_n m^{-1/n}$, and for $m \leq i \leq 2^n m$ let $B_i$ be the ball with center $x_i$ and radius $\frac{1}{2} h_n (2i^{-1/n} - m^{-1/n})$. Show that the $2^n m$ balls $B_i$ are pairwise disjoint, and evaluate the measure of their union, making use of the Euler–Maclaurin formula.)