2 HILLEL FURSTENBERG of points {0, 1 n , 2 n , . . . , 1} if n 1 ε despite the fact that [0, 1] is a continuum and {0, 1 n , 2 n , . . . , 1} is a discrete set. Now we are ready to define mini- and micro-sets. Note that this terminology is not universally accepted, so some other books may use different names for the same objects. We will always assume that our starting set A is a compact subset of Rn contained in the unit cube Q = [0, 1]n. Definition. A mini-set of A is any set of the kind (λA + u) ∩ Q where λ ≥ 1 and u ∈ Rn are such that Q ⊂ λQ + u. The set A A mini-set of A Figure 2. A mini-set of A is the scaled intersection of A with a small square window (enlarged on the right). A micro-set of A is the limit of any sequence An of mini-sets of A in the Hausdorff metric. Informally, a mini-set is what you see if you look at a small portion of a set through a magnifying glass. The condition Q ⊂ λQ + u is needed to ensure that a mini-set of a mini-set of A is again a mini-set of A. Note also that the scales λn and the shifts un corresponding to the mini-sets An in the definition of a micro-set do not need to be related in any way.1 A lot of interesting micro-sets can be obtained by zooming at a single point like at the picture below, but we also have an option of moving our window around The set A A micro-set of A Mini-sets of A Figure 3. A micro-set of a classical set is a union of several points. simultaneously with the scale reduction. The most interesting micro-sets are those that correspond to the sequences of mini-sets An = (λnA + un) ∩ Q with λn → +∞. For all objects studied in classical geometry, they are essentially flat, i.e., either planes or unions of a few planes as 1 Note also that the set of closed subsets of the closed unit cube Q endowed with the Haus- dorff metric is compact (Blaschke theorem), so any sequence of mini-sets contains a convergent subsequence.

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