By Yomdin Y., Comte G.

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Then: and ∞ i=1 αiβ < ∞}. dime (A) = inf{β; i=1 For example, for A = {1, 1 1 2a , 3a , . }, αi ∼ 1 ia+1 , and hence dime (A) = 1 . a+1 This theorem allows us to compute again dime (C 13 ): for 2n−1 i < 2n , 1 αi = ( )n , thus 3 ∞ ∞ 1 2 ( β )i−1 , and this sum is convergent if and 3 3 i=1 i=1 log(2) . 10. Let A ⊂ Rn be a bounded subset. For a given β > 0, points x1 , . . , xp ∈ A, and a connected tree T with vertices xi , ρβ (x1 , . . , xp , T ) is the sum |e|β , where for an edge e in T connecting xi and xj , |e| = e∈T d(xi ; xj ).

4) For a convex subset A in Rn , Vi (A) = Wi (A, B n ), where Wi denotes the Minkowski mixed volume, and B n ⊂ Rn is the unit ball. ) (5) Vi (A) are invariants of the isometries of Rn . (6) Homogeneity property: For λ ∈ R, Vi (λA) = λi Vi (A). (7) Vi (A ∪ B) ¯ = ∅ we have the equality. Vi (A) + Vi (B). If A¯ ∩ B (8) Inductive formula for variations: Vi (A) = c(n, i, j) n−j ¯n P¯ ∈G Vi−j (A ∩ P¯ ) dP¯ . In this formula, A ∩ P¯ is a subset of P¯ = Rn−j , thus we have in this ¯ dQ. ¯ V0 (A ∩ P¯ ∩ Q) formula: Vi−j (A ∩ P¯ ) = c(n − j, i − j) ¯ G ¯ i−j P¯ ⊃Q∈ n−j 36 3 Multidimensional Variations It is an exercise to compute the constant c(n, i, j); for instance when i = n−1 and j = 1 we have: Vn−1 (A) = c(n, n − 1, 1) n−1 ¯n P¯ ∈G Vn−2 (A ∩ P¯ ) dP¯ .

If we consider in addition the function x → ex , we obtain the so-called Log-Analytic structure, denoted S(Ran,exp ). ) ⊂ S(Ran ) ⊂ S(Ran,exp ). ) and the structure consisting of all semialgebraic sets). See also [Shi], for an interesting and slightly diﬀerent (actually, a more general) viewpoint. 17. We will say that a set belonging to an analytic-geometric category or to an o-minimal structure is a tame set (see [Tei 2]). 18. Every tame set has the local Gabrielov property (the global Gabrielov property if the set is in an o-minimal structure).