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Let $V$ be a bounded open subset of $\mathbb{R}^n.$ Is $B(0,1)$ the union of countably many (pairwise disjoint) rescalings/translates of $V$?

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Let $V$ be a bounded open subset of $\mathbb{R}^n,$ and define $B(x,r)$ to be the open ball with centre $x$ and radius $r.$

True or false: $B(0,1)$ is the union of countably many pairwise disjoint rescalings/translates of $V$.

In other words, if $a,\in\mathbb{R}, b,\in\mathbb{R}^n$ then define $V_{a,b} := \{ax+b: x\in V\}.$ The question then is whether or not there exist sequences $(a_n)_n$ and $(b_n)_n$ such that $V_{a_k, b_k} \cap V_{a_j,b_j} = \emptyset\ $ if $ j\neq k $ and

$$ \bigcup_{n\in\mathbb{N},\ x\in V} V_{a_n,b_n} = B(0,1).$$

Perhaps the result is false. For example, maybe it cannot be done if we define $V$ as follows:

Let $(x_n)_{n\in\mathbb{N}}$ be an enumeration of $\mathbb{Q}^n\cap B(0,1),$ let $y_n=3^{-n},$ and define $V =\bigcup_{n\in\mathbb{N}} B(x_n,y_n).$

The hard part of this question seems to be a geometric one: how can we show that we can entirely fill up $B(0,1)$ for any possible shape that $V$ can be - or alternatively, is there a counter-example?


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