TY - JOUR
AU - Zindulka,, Ondřej
AB - Abstract If X is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure μ on X there is a set N ⊆ X such that μ(N) > 0, an ultrametric space Z and a Lipschitz bijection ϕ : N → Z whose inverse is nearly Lipschitz, i.e., β-Hölder for all β < 1. As an application it is shown that a Borel set in a Euclidean space maps onto [0, 1]n by a nearly Lipschitz map if and only if it cannot be covered by countably many sets of Hausdorff dimension strictly below n. The argument extends to analytic metric spaces satisfying the mild condition. Further generalization replaces cubes with self-similar sets, nearly Lipschitz maps with nearly Hölder maps and integer dimension with arbitrary finite dimension. 1 Introduction It is easy to prove that every compact set |$K\subseteq \mathbb{R}$| of real numbers with positive Lebesque measure can be mapped onto the interval [0, 1] by a Lipschitz map. In [17], Miklós Laczkovich asked if this remains true in higher dimensions. In more detail, if it is true that for every natural number n > 1 and every compact set |$K\subseteq \mathbb{R}^{n}$| with positive n-dimensional Lebesgue measure there is a Lipschitz mapping f : K → [0, 1]n onto the n-dimensional cube. This question turned to be very difficult. So far the (affirmative) answer was found only for n = 2 (by David Preiss, published years later in [2]). Vitushkin, Ivanov, and Melnikov [27] (see also [15]) constructed an example that shows that no generalization beyond the Laczkovich’s question is possible: a compact set |$K\subseteq \mathbb{R}^{2}$| with positive linear measure (i.e., the 1-dimensional Hausdorff measure) that cannot be mapped onto a segment by a Lipschitz map. Let n be a positive natural number and let us denote the n-dimensional Hausdorff measure by |$\mathcal{H}^{n}$| and the Hausdorff dimension by |$\operatorname{\dim _{\mathsf{H}}}$|⁠. Using recent results on monotone metric spaces [28] and ultrametric spaces [19] (see below) Keleti, Máthé, and Zindulka [16] proved that if the assumption |$\mathcal{H}^{n}(K)>0$| is strengthened to |$\operatorname{\dim _{\mathsf{H}}} K>n$|⁠, then K can be mapped by a Lipschitz map onto [0, 1]n for any analytic metric space K. From what have been said, it is clear that we do not have a complete understanding of what condition akin to |$\mathcal{H}^{n}(K)>0$| is equivalent to the existence of a Lipschitz mapping of K onto [0, 1]n, what remains true about mapping of K onto [0, 1]n if we merely suppose that |$\mathcal{H}^{n}(K)>0$|⁠. The present paper deals with the latter question and provides a partial answer. We build upon ideas presented in [19] and [16]. In [19], Mendel and Naor prove that a compact metric space contains a large (with respect to Hausdorff dimension) subset with a rather simple metric structure—it is, up to bi-Lipschitz homeomorphism, ultrametric. We also construct a large subset with a simple structure, the “large” and “simple” notions, however, are a bit different. In more detail, we prove in Section 2 a theorem about spaces satisfying a mild doubling condition henceforth termed non-exploding spaces. Roughly speaking, the condition requires that the number of balls of radius r needed to cover X does not increase too fast with decreasing r, cf. Definition 2.1. The following is a simplified version of Theorem 2.6. Theorem. Let X be a non-exploding analytic metric space and let μ be a finite Borel measure on X. Then there is a set N ⊆ X such that μ(N) > 0, an ultrametric space Z and a Lipschitz bijection ϕ : N → Z whose inverse is β-Hölder for all β < 1. Then, based upon this result, we show that if a non-exploding analytic metric space has positive n-dimensional Hausdorff measure, then there is a mapping of X onto [0, 1]n that is as close to Lipschitz as it gets. (We know from the Vitushkin’s example that it may fail to be Lipschitz.) Theorem. Let X be an analytic non-exploding space and |$n\in \mathbb{N}$|⁠. If |$\mathcal{H}^{n}(X)>0$|⁠, then there is a mapping f : X → [0, 1]n onto [0, 1]n that is β-Hölder for every β < 1. This is proved in Section 4. The full strength of the mapping theorem is stated in Theorem 4.3 and its Corollaries 4.4 and 4.5. Once we have this mapping result, we may ask if the sufficient condition on the Hausdorff measure is also necessary. And it turns out that it is not, nevertheless there is a simple, natural condition on X involving Hausdorff dimension that is necessary and also sufficient. We discuss this in Section 3. Further generalization replaces cubes with self-similar sets, nearly Lipschitz maps with nearly Hölder maps, and integer dimension with arbitrary finite dimension. The last Section 5 contains remarks and presents some questions and problems. The results of this paper already found an application. Namely, Balka, Elekes, and Máthé use them in [6] to prove the following theorem on a prevalent behavior of continuous functions. (The dimensions involved are the Hausdorff and packing dimensions of sets and measures.) Theorem ([6]). Let K be a non-exploding compact metric space and μ a continuous, finite Borel measure on K. Let n be a positive integer. Then for almost every continuous function on K (in the sense of Christensen’s [8] Haar measure zero) with values in |$\mathbb{R}^{n}$| there is an open set |$U_{f}\subseteq \mathbb{R}^{n}$| such that |$\mu \left (f^{-1}\left (U_{f}\right )\right )=\mu (K)$| and for all y ∈ Uf $$ \operatorname{\dim_{\mathsf{H}}} f^{-1}(y)\geqslant\operatorname{\dim_{\mathsf{H}}}\mu\quad\textrm{ and }\quad\operatorname{\dim_{\mathsf{P}}} f^{-1}(y)\geqslant\operatorname{\dim_{\mathsf{P}}}\mu. $$ This theorem generalizes a number of previous results and, as far as I know, it is the first theorem of its kind proved in such a general context. All spaces we work with are separable metric spaces. Recall that a metric space is analytic if it is a continuous image of a complete metric space (or, equivalently, of the irrational numbers, or equivalently, a Suslin set in a complete metric space). A continuous image of an analytic space is analytic. Every analytic space is separable. Some of the common notation includes B(x, r) for the closed ball centered at x, with radius r; diamE for the diameter of a set E in a metric space; dist(A, B) the (lower) distance of two sets A, B; n, m are generic symbols for positive integers. |$\mathbb{R}$| and |$\mathbb{Z}$| have the usual meaning; ω stands for the set of natural numbers including zero. 2ω denotes the set of binary sequences; it is also a compact topological space homeomorphic to the standard Cantor ternary set and a topological group. Likewise, ωω denotes the set of all sequences of natural numbers, i.e., the maps f : ω → ω. 2 Large Nearly Ultrametric Sets In [19], Mendel and Naor proved that every analytic metric space X contains large ultrametric-like subspaces. In more detail, for every ε > 0 there is a subset Y ⊆ X with the following two properties: (1) Hausdorff dimension of Y is large: |$\operatorname{\dim _{\mathsf{H}}} Y\geqslant \operatorname{\dim _{\mathsf{H}}} X-\varepsilon $|⁠, (2) the metric structure of Y is ultrametric-like: there is a bijection f : Y → U onto an ultrametric space U such that both f and its inverse are Lipschitz. Though it is not clear at first glance how this powerful result is related to mapping metric spaces onto cubes, it is one of two crucial ingredients of the theorems of [16] mentioned in the previous paragraph and its spirit, as we shall see, is also important for the present paper. In this section we attempt to prove a theorem similar to that of Mendel and Naor: our goal is to find, just like in the theorem, within an analytic metric space X a large ultrametric-like set Y. We want a bit more than (1): given a finite Borel measure on X, our set will have to have positive measure. In order to achieve that, we have to sacrifice some of (2): our set will be still ultrametric-like, but not quite as much as in (2). We term the notion nearly ultrametric; it is introduced in Definition 2.4 below. We do not succeed completely: our proof only works within a framework of analytic spaces that are subject to a growth condition similar to the doubling condition, but much weaker, the so-called non-exploding spaces. The result is stated in Theorem 2.6. Non-exploding spaces We first discuss the doubling-like condition. Recall that a metric space X is doubling if there is a number Q such that every ball in the space X can be covered by at most Q many balls of halved radii. This notion and equivalent or similar notions have been defined, investigated, and used throughout the literature. We may generalize the notion as follows. Suppose that the number Q is not fixed but may increase as the radius of the ball in question decreases; but it may not increase very fast. In more detail: Definition 2.1. Let X be a metric space. If there is a function |$Q:(0,\infty )\to \mathbb{R}$| such that every closed ball in X of radius r > 0 is covered by at most Q(r) many closed balls of radius r/2 and such that $$\begin{align} \lim_{r \to 0}\frac{\log Q(r)}{\log r}=0 , \end{align}$$ (NE) we call the metric space X non-exploding. The term was coined by Tamás Keleti. Needless to say that every doubling metric space and in particular every subset of a Euclidean space is non-exploding. Nearly Lipschitz maps We will be frequently making use of the notion of a nearly Lipschitz map that was introduced in [28]. We present two of the several equivalent definitions. Recall that, given β > 0, a mapping f : (X, dX) → (Y, dY) is β-Hölder if there is an ε > 0 and a constant C such that if |$d_{X}(x,y)\leqslant \varepsilon $|⁠, then |$d_{Y}\left (f(x),f(y)\right )\leqslant C\,d_{X}(x,y)^{\beta }$|⁠. (Note that we deviate slightly from the usual definition by introducing ε, but if Y is bounded, the definitions are equivalent and, moreover, since we are interested in low-scale behavior, we may always suppose that Y is bounded.) Definition 2.2. ([28]) A mapping f : (X, dX) → (Y, dY) between metric spaces is termed nearly Lipschitz if f is β-Hölder for all β < 1. Proposition 2.3. ([28]) A mapping f : (X, dX) → (Y, dY) is nearly Lipschitz if and only if there is a function |$h:(0,\infty )\to (0,\infty )$| such that |$\varliminf _{r\to 0}\frac{\log h(r)}{\log r}\geqslant 1$| and $$\begin{align} d_{Y}\left(f(x),f(\,y)\right)\leqslant h\left(d_{X}(x,y)\right), \quad x,y\in X. \end{align}$$ (1) Proof. Suppose that f : X → Y is nearly Lipschitz. Then there is a decreasing sequence δn → 0 and a sequence of constants Cn such that $$ d_{X}(x,y)\leqslant\delta_{n} \implies d_{Y}\left(\,f(x),f(y)\right)\leqslant C_{n} d_{X}(x,y)^{1-1/n}. $$ We may also suppose that |$\delta _{n}\leqslant C_{n}^{-n}$| so that |$\delta _{n}^{-1/n}$| is an upper estimate of Cn. Define the function h by $$ h(r)=r^{1-2/n}\ \textrm{if}\ r\in\big[{\delta_{n+1},\delta_{n}}\big). $$ Since |$\log h(r)/\log r=1-2/n$| on the entire interval |$\left [\delta _{n+1},\delta _{n}\right )$|⁠, we have |$ \varliminf _{r\to 0}\frac{\log h(r)}{\log r}=\varliminf _{n\to \infty }1-\frac 2n=1. $| If |$\delta _{n+1}\leqslant d_{X}(x,y)<\delta _{n}$|⁠, then $$ d_{Y}\left(\,f(x),f(\,y)\right)\leqslant\delta_{n}^{-1/n}d_{X}(x,y)^{1-1/n}\leqslant d_{X}(x,y)^{1-2/n}=h\left(d_{X}(x,y)\right)\!, $$ which proves (1). The reverse implication is straightforward. We will say that the metric spaces X, Y are nearly Lipschitz equivalent if there is a bijective mapping f : X → Y such that both f and its inverse are nearly Lipschitz. Nearly ultrametric spaces Recall that a metric space (X, d) is ultrametric if any triple x, y, z ∈ X of points satisfies |$d(x,z)\leqslant \max \left \{d(x,y),d(y,z)\right \}$|⁠. Definition 2.4. A metric space X is nearly ultrametric if it is nearly Lipschitz equivalent to an ultrametric space. Proposition 2.5. Let X be a metric space. The following are equivalent: (i) X is nearly ultrametric, (ii) there is a nearly Lipschitz bijection f : X → Y onto an ultrametric space with Lipschitz inverse, (iii) there is a Lipschitz bijection f : X → Y onto an ultrametric space with nearly Lipschitz inverse. Proof. Of course it is enough to prove (i)⇒(ii) and (i)⇒(iii). Using Proposition 2.3 there are functions g, h such that |$\varliminf _{r\to 0}\frac{\log h(r)}{\log r}\geqslant 1$| and |$\varliminf _{r\to 0}\frac{\log g(r)}{\log r}\geqslant 1$| and such that $$ d_{X}(x,y)\leqslant g\left(d_{Y}\left(\,f(x),f(y)\right)\right)\leqslant g\circ h\left(d_{X}(x,y)\right). $$ (The latter inequality holds if g is nondecreasing, but we may suppose that.) Define a new metric on Y by |$d_{Y}^{\prime }=g\circ d_{Y}$|⁠. Since dY is an ultrametric, so is |$d_{Y}^{\prime }$| and it is easy to verify that the mapping |$f:(X,d_{X})\to (Y,d_{Y}^{\prime })$| is nearly Lipschitz and has Lipschitz inverse. This proves (i)⇒(ii) and (i)⇒(iii) is proved likewise. We have enough to state the first summit of the paper. Theorem 2.6. Let X be a non-exploding analytic metric space and let μ be a finite Borel measure on X. Then for every ε > 0 there is a compact nearly ultrametric set C ⊆ X such that μ(X ∖ C) < ε. The rest of this section is devoted to the proof of this theorem. It has two stages: we first prove a very particular case and then reduce the theorem to this particular case. The infinite-dimensional torus We will prove the theorem first for the infinite-dimensional torus, i.e., the compact group |$\left (\mathbb{R}/\mathbb{Z}\right )^{\omega }$| equipped with a special metric, and its Haar measure. We will identify |$\mathbb{R}/\mathbb{Z}$| with the interval [0, 1). The group operation on [0, 1) is of course addition modulo 1. For x, y ∈ [0, 1) the distance from x to y is given by|$d(x,y)= \left \| x-y \right \|$|⁠, where |$\left \|z\right \|=\min \left \{|z|,|1-z|\right \}$|⁠, i.e., [0, 1) is a circle and |$\lVert x-y\rVert $| is the length of the shorter of the two arcs between x and y. We also consider the Lebesgue measure on [0, 1); let us denote it by |$\mathcal{L}$|⁠. Let |$\mathbb{T}=[0,1)^{\omega }$| and denote the probability Haar measure on |$\mathbb{T}$| by |$\lambda _{\mathbb{T}}$| (which is clearly obtained as a product measure from |$\mathcal{L}$|⁠). Now let G ∈ ωω be a sequence that is not eventually zero. Such a function defines a partition of |$\omega =I_{0}\cup I_{1}\cup I_{2}\cup \dots $| into disjoint consecutive intervals such that the length of each In is G(n). (Some of the intervals may be empty.) For each n and k ∈ In let rk = 2−n. Clearly rk ↘ 0. Now define the metric on |$\mathbb{T}$| as follows: $$ d_{G}(x,y)=\sup_{k\in\omega}r_{k}\left\|x_{k}-y_{k}\right\|. $$ It is easy to check that dG is an invariant metric on |$\mathbb{T}$|⁠. It is clear that if G assumes large values, then the diameters of the coordinate circles decrease slowly, and vice versa. We impose the following growth condition upon G. From now on we define |$\log 0=0$|⁠. Definition 2.7. We call a sequence G ∈ ωωslow if G(n) > 0 for infinitely many n and |$\log G(n)/n\to 0$|⁠. Lemma 2.8. If G is slow, then for every ε > 0 there is a compact nearly ultrametric set |$C\subseteq (\mathbb{T},d_{G})$| such that |$\lambda _{\mathbb{T}}(C)>1-\varepsilon $|⁠. Proof. We first construct the set and then prove that it has the required properties. Begin with defining two sequences of positive real numbers |$\left \langle a_{m}\right \rangle ,\left \langle b_{k}\right \rangle $|⁠: let $$ a_{m}=\frac{1}{2\left(m+1\right)^{2}},\qquad b_{k}=\frac{\varepsilon}{4\left(n+1\right)^{2}G(n)}, $$ where n is the unique number such that k ∈ In. Clearly $$\begin{align} \sum_{m\in\omega}a_{m}<1,\qquad \sum_{k\in\omega}b_{k}<\varepsilon/2. \end{align}$$ (2) We now set up, for each k, a Cantor-like ternary set Ck ⊆ [0, 1). Let |$2^{<\omega }=\bigcup _{n\in \omega }2^{n}$| be the binary tree consisting of all {0, 1}-valued finite sequences. Build recursively a family of closed intervals |$\left \{J_{s}:s\in 2^{<\omega }\right \}$| as follows. Let |$J_{\emptyset }=\left [0,1-b_{k}\right ]$| and if s ∈ 2<ω and Js is constructed, define |$J_{s\!\frown\;\;0}$| and |$J_{s\!\frown\;\;1}$| subject to the following conditions: (a) |$J_{s\!\frown\;\;0}$| and |$J_{s\!\frown\;\;1}$| are disjoint equally long subintervals of Js, (b) the left endpoints of Js and |$J_{s\!\frown\;\;0}$| coincide, (c) the right endpoints of Js and |$J_{s\!\frown\;\;1}$| coincide, (d) the gap between |$J_{s\!\frown\;\;0}$| and |$J_{s\!\frown\;\;1}$| is 2−|s|a|s|bk. Conditions (2) ensure that the construction is possible. Let $$ C_{k}=\bigcap_{p\in\omega}\bigcup_{s\in 2^{p}}J_{s} $$ be the resulting ternary set. Let $$ C=\prod_{k\in\omega}C_{k}\subseteq\mathbb{T}. $$ The first property we notice is |$\lambda _{\mathbb{T}}(C)>1-\varepsilon $|⁠. Indeed, Ck is omitting a set of Lebesgue measure exactly |$b_{k}\left (1+\sum _{m\in \omega }a_{n}\right )<2b_{k}$|⁠. Hence |$\mathcal{L}\left (C_{k}\right )>1-2b_{k}$|⁠. Therefore $$ \lambda_{\mathbb{T}}(C)=\prod_{k}\left(1-2b_{k}\right)\geqslant 1-2\sum_{k} b_{k}>1-\varepsilon. $$ We claim that C is also nearly ultrametric. To prove it, consider the Cantor cube 2ω and provide it with the usual least difference metric: |$\rho (x,y)=2^{-\left \lVert x\wedge y\right \rVert }$| if |$x\neq y$|⁠, ρ(x, x) = 0. (Here and later, x ∧ y denotes the initial segment common to x and y.) For x ∈ 2ω denote by |$\widehat x$| the unique point of Ck coded by x. Now consider the power |$\left (2^{\omega }\right )^{\omega }$| of the Cantor set 2ω and equip it with the metric $$ \rho_{G}(x,y)=\sup_{k\in\omega}r_{k}\rho\left(x_{k},y_{k}\right). $$ For |$x\in \left (2^{\omega }\right )^{\omega }$| let |$\widehat x=\left \langle \widehat x_{k}:k\in \omega \right \rangle $|⁠. The mapping |$x\mapsto \widehat x$| is obviously bijective. It is also Lipschitz: let |$x,y\in \left (2^{\omega }\right )^{\omega }$| and let k ∈ ω. Set s = xk ∧ yk. Then |$\left \lVert \widehat x_{k}-\widehat y_{k}\right \rVert \leqslant \mathcal{L}(J_{s})\leq 2^{-|s|}=\rho (x_{k},y_{k})$|⁠. Since this is true for every coordinate, we have |$d_{G}\left (\widehat x,\widehat y\right )\leqslant \rho _{G}(x,y)$|⁠. We now prove that the inverse |$\widehat x\mapsto x$| of the map is nearly Lipschitz. In view of Proposition 2.3 it is thus enough to proceed as follows: let j ∈ ω and suppose ρG(x, y) = 2−j. There is n and k ∈ In such that |$r_{k}\rho \left (x_{k},y_{k}\right )=2^{-j}$|⁠. Put s = xk ∧ yk and p = |s|. Since rk = 2−n, we have 2−n2−p = 2−j, i.e., n + p = j. Estimate |$d_{G}\left (\widehat x,\widehat y\right )$|⁠: $$ d_{G}\left(\widehat x,\widehat y\right)\geqslant r_{k}\left\lVert\widehat x_{k}-\widehat y_{k}\right\rVert \geqslant r_{k}\operatorname{dist}\left(J_{s\!\frown\;\;0},J_{s\!\frown\;\;1}\right) =r_{k}b_{k}2^{-p}a_{p}=2^{-j}b_{k}a_{p}. $$ Therefore (employing definitions of am and bk) $$\begin{align*} \frac{\log d_{G}\left(\widehat x,\widehat y\right)}{\log\rho_{G}(x,y)}\leqslant&\ \frac{j-\log a_{p}-\log b_{k}}{j} \\ =&\ \frac{j+2\log(\,p+1)+1+2\log(n+1)+2+\log G(n)-\log\varepsilon}{j}. \end{align*}$$ Let |$\tilde G(n)=\max _{i\leqslant n}G(i)$|⁠. It is easy to check that since G is slow, so is |$\tilde G$|⁠. Since n + p = j, we conclude that $$\begin{align*} \limsup_{j\to\infty}\sup_{\rho_{G}(x,y)=2^{-j}} \frac{\log d_{G}\left(\,\widehat x,\widehat y\right)}{\log\rho_{G}(x,y)} \leqslant \limsup_{j\to\infty} \frac{j+4\log(\ j+1)+3+\log\tilde G(\ j)-\log\varepsilon}{j}=1, \end{align*}$$ which is enough. In summary, the mapping |$x\mapsto \widehat x$| is a nearly Lipschitz equivalence of C and |$\left ((2^{\omega })^{\omega },\rho _{G}\right )$|⁠. Since |$\left ((2^{\omega }\right )^{\omega },\rho _{G})$| is an ultrametric space, it follows that C is nearly ultrametric. We now extend the statement to all measures on |$\mathbb{T}$|⁠. Proposition 2.9. Let μ be a finite Borel measure on |$\mathbb{T}$| and ε > 0. If G is slow, then there is a compact nearly ultrametric set |$K\subseteq (\mathbb{T},d_{G})$| such that |$\mu \left (\mathbb{T}\setminus K\right )<\varepsilon $|⁠. Proof. Assume without loss of generality that |$\mu \left (\mathbb{T}\right )=1$|⁠. Let |$C\subseteq \mathbb{T}$| be the nearly ultrametric set from Lemma 2.8. Let |$A=\left \{(x,y):x+y\in C\right \}$|⁠. By Fubini Theorem, |$\left (\lambda _{\mathbb{T}}\times \mu \right )(A)=\int _{\mathbb{T}}\lambda _{\mathbb{T}}(C-x)\;\mathrm{d}\mu =\lambda _{\mathbb{T}}(C)\mu (\mathbb{T})$| and at the same time |$\left (\lambda _{\mathbb{T}}\times \mu \right )(A) =\int _{\mathbb{T}}\mu (C-x)\;\mathrm{d}\mu $|⁠. Thus $$ \int_{\mathbb{T}}\mu(C-x)\;\mathrm{d}\mu=\lambda_{\mathbb{T}}(C)\mu(\mathbb{T})=\lambda_{\mathbb{T}}(C)>1-\varepsilon $$ and consequently there is x such that μ(C − x) > 1 −ε. Since C is nearly ultrametric and dG is a shift invariant, it follows that C − x is also nearly ultrametric. (Observant reader may have noticed that we actually reproved the famous Christensen characterization [8] of Haar measure zero sets in locally compact groups in a slightly stronger setting. We could alternatively use the Christensen’s theorem as a black box.) Assouad’s embedding revisited To prove Theorem 2.6, it is enough to reduce it to the case covered by the above proposition. In order to do so we prove the following theorem. It is akin to classical embedding theorems of Aharoni [1] and Assouad [4, 5]. We will reiterate some of the ideas of their proofs, in particular those that appear in [5]. Theorem 2.10. For every compact non-exploding metric space X there is a slow G ∈ ωω and a bi-Lipschitz embedding |$f:X\hookrightarrow \left (\mathbb{T},d_{G}\right )$|⁠. Construction Let X be a metric space. The symbol B(x, r) denotes, as usual, a closed ball with center x and radius r. Fix ε > 0. Suppose S ⊆ X is a maximal ε-separated set in X (i.e., d(s, s′) > ε for distinct s, s′∈ S). Let I be a finite set and N ∈ ω its cardinality. Suppose that for every x ∈ X we have $$\begin{align} \left\lvert S\cap B(x,8\varepsilon)\right\rvert\leqslant N. \end{align}$$ (3) We employ Assouad’s [5, Lemme 2.4] that claims that if (3) holds, then there is a coloring χ : S → I such that if d(s, s′) ⩽ 8ε, then |$\chi (s)\neq \chi (s^{\prime })$|⁠. Fix such a coloring χ and define, for each j ∈ I, a function |$\phi _{j}:X\to \mathbb{R}$| as follows. Let x ∈ X. The set |$\chi ^{-1}(j)\cap B\left (x,\frac 32\varepsilon \right )$| has at most one point—otherwise there would be two points in S with the same color and within distance 3ε. If there is a (unique) |$s\in \chi ^{-1}(j)\cap B\left (x,\frac 32\varepsilon \right )$|⁠, let ϕj(x) = d(x, s), otherwise let |$\phi _{j}(x)=\frac 32\varepsilon $|⁠. Lemma 2.11. |$\forall x,y\in X\ \forall j\in I\ \left \lvert \phi _{j}(x)-\phi _{j}(y)\right \rvert \leqslant d(x,y)$| Proof. Clearly |$\phi _{j}(x)=\min \left \{\underline{d}\left (x,\chi ^{-1}\left (j\right )\right ),\tfrac 32\varepsilon \right \}$|⁠, where |$\underline{d}$| denotes the lower distance of a point from a set. The inequality $$ \underline{d}\left(x,\chi^{-1}\left(j\right)\right)\leqslant d(x,y)+\underline{d}\left(y,\chi^{-1}(j)\right) $$ is immediate from the triangle inequality. Hence $$ \phi_{j}(x)\leqslant\min\left\{d(x,y)+\underline{d}\left(y,\chi^{-1}(\,j\,)\right),\tfrac32\varepsilon\right\} \leqslant d(x,y)+\phi_{j}(\,y) $$ as required. Lemma 2.12. If |$\frac 52\varepsilon < d(x,y)\leq 5\varepsilon $|⁠, then there is j ∈ I such that $$ \left\lvert\phi_{j}(y)-\phi_{j}(x)\right\rvert\geqslant\tfrac1{10}d(x,y). $$ Proof. Since S is maximal ε-separated, there is s ∈ S ∩ B(x, ε). Let j = χ(s). For every |$s^{\prime}\in S\cap B\left (y,\frac 32\varepsilon \right )$| we have $$ d\left(s,s^{\prime}\right)\leqslant d(s,x)+d(x,y)+d\left(y,s^{\prime}\right)\leqslant\varepsilon+5\varepsilon+\tfrac32\varepsilon<8\varepsilon. $$ At the same time $$ d\left(s,s^{\prime}\right)\geqslant d(x,y)-d(s,x)-d\left(y,s^{\prime}\right)>\tfrac52\varepsilon-\varepsilon-\tfrac32\varepsilon=0, $$ which proves |$s\neq s^{\prime }$| and thus |$\chi \left (s^{\prime }\right )\neq \chi (s)=j$|⁠. It follows that |$\phi _{j}(y)=\frac 32\varepsilon $| and consequently $$ \left\lvert\phi_{j}(y)-\phi_{j}(x)\right\rvert\geqslant\tfrac32\varepsilon-d(x,s)\geqslant\tfrac12\varepsilon= \tfrac1{10}\,5\varepsilon\geqslant\tfrac1{10}d(x,y). $$ Proof of Theorem 2.10 Since X is compact, we may assume that its diameter is bounded by 1. For every n let εn = 2−n. Let Sn ⊆ X be a maximal εn-separated set and let G(n) ∈ ω be a minimal number such that $$ \forall x\in X\quad\left\lvert S_{n}\cap B(x,8\varepsilon_{n})\right\rvert\leqslant G(n). $$ Claim.G, defined as above, is slow. Indeed, letting $$ Q(\varepsilon)=\min\left\{Q\in\omega: \forall x\in X\ \exists \left\{x_{i}:i < Q\right\}\ B(x,\varepsilon)\subseteq\bigcup_{i < Q}B\left(x_{i},\varepsilon/2\right)\right\} $$ it is easy to check that |$G(n)\leqslant Q\left (8\varepsilon _{n}\right ) Q(4\varepsilon _{n}) Q(2\varepsilon _{n}) Q(\varepsilon _{n})$| and thus $$ \lim_{n\to\infty}\frac{\log G(n)}{n}\leqslant \lim_{n\to\infty} \frac{\log Q(8\varepsilon_{n})+\log Q(4\varepsilon_{n})+\log Q(2\varepsilon_{n})+\log Q(\varepsilon_{n})}{\log\varepsilon_{n}} =0, $$ because X is non-exploding. Let |$\omega =I_{0}\cup I_{1}\cup I_{2}\cup \dots $| be the partition of ω into disjoint consecutive intervals such that the length of each In is G(n), as discussed earlier, and let χn : Sn → In be the corresponding coloring. For every j ∈ In let |$\phi _{j}:X\to \mathbb{R}$| be the mapping constructed above for χ = χn, S = Sn, ε =εn. It is clear that |$\phi _{j}(X)\subseteq \left [0,\frac 32\varepsilon _{n}\right ]$|⁠. Thus the mapping ϕ on X defined by |$\phi (x)=\left \langle \frac 13\phi _{j}(x):j\in \omega \right \rangle $| maps X into the cube |$\prod _{n\in \omega }\left [0,2^{-n-1}\right ]^{I_{n}}$|⁠. Equip this cube with the supremum metric. Now Lemma 2.11 proves that ϕ is Lipschitz and Lemma 2.12 proves that ϕ−1 is Lipschitz. Overall, |$\phi :X\hookrightarrow \prod _{n\in \omega }\left [0,2^{-n-1}\right ]^{I_{n}}$| is a bi-Lipschitz embedding. For each n and j ∈ In let ψj be the linear function that maps |$\left [0,2^{-n-1}\right ]$| onto |$\left [0,\frac 12\right ]$|⁠. The mapping |$\psi =\left \langle \psi _{j}:j\in \omega \right \rangle $| is clearly an isometric embedding of |$\prod _{n\in \omega }\left [0,2^{-n-1}\right ]^{I_{n}}$| into |$\left (\mathbb{T},d_{G}\right )$|⁠. Thus f = ψ ∘ ϕ is the required bi-Lipschitz embedding |$f:X\hookrightarrow \left (\mathbb{T},d_{G}\right )$|⁠. Proof of Theorem 2.6 The proof of the main theorem is now trivial: let μ be the finite Borel measure on X; we may suppose μ(X) = 1. Since X is analytic, there is a compact set C ⊆ X such that μ(C) > 1 −ε/2. By the above embedding Theorem 2.10 there is a slow sequence G ∈ ωω and a bi-Lipschitz embedding |$C\hookrightarrow \left (\mathbb{T},d_{G}\right )$|⁠. Let ν be the image measure of the restriction of μ to the set C. By Proposition 2.9 there is a nearly ultrametric set |$K\subseteq \mathbb{T}$| such that |$\nu (K)>\nu \left (f(C)\right )-\varepsilon /2=\mu (C)-\varepsilon /2>1-\varepsilon $|⁠. The desired set is |$f^{-1}\left (f(C)\cap K\right )$|⁠. 3 Indecomposability Given n ∈ ω, if the n-dimensional Hausdorff measure of a metric space X is positive, then its Hausdorff dimension is at least n, but not necessarily vice versa. We will study a condition between the two, whose importance lies in the fact that it is sufficient and also necessary for X to be mapped onto an n-dimensional ball by a nearly Lipschitz mapping. The main result of this section is Theorem 3.3. We have to go through some preliminary material first. Hausdorff functions Recall that a right-continuous, nondecreasing function |$h:[0,\infty )\to [0,\infty )$| with h(0) = 0 is called a gauge or a Hausdorff function. Recall that a gauge is doubling if there is a constant C such that |$h(2r)\leqslant Ch(r)$| for all r > 0. For a gauge h define $$ \operatorname{ord} h=\varliminf_{r\to0}\frac{\log h(r)}{\log r}. $$ Hausdorff measures Recall that given a gauge g, the Hausdorff measure|$\mathcal{H}^{g}$| on a metric space X is defined thus: for each δ > 0 and E ⊆ X set $$\begin{align} \mathcal{H}^{g}_{\delta}(E)=\inf\sum_{n}g(\operatorname{diam} E_{n}), \end{align}$$ (4) where the infimum is taken over all finite or countable covers {En} of E by sets of diameter at most δ, and put $$ \mathcal{H}^{g}(E)=\sup_{\delta>0}\mathcal{H}^{g}_{\delta}(E). $$ The basic properties of |$\mathcal{H}^{g}$| are well-known. It is an outer measure and its restriction to Borel sets is a Gδ-regular Borel measure in X. General references: [10, 18, 24]. We shall need the following theorem of Howroyd [12] that generalizes earlier results of Besicovitch [7] and Davies [9]. Theorem 3.1. (Howroyd [12]) Let X be an analytic metric space and g a doubling gauge. If |$\mathcal{H}^{g}(X)>0$|⁠, then there is a compact set K ⊆ X such that |$0<\mathcal{H}^{g}(K)<\infty $|⁠. The particular case of Hausdorff measure when the gauge is given by g(x) = xs for a fixed s > 0 is of major importance. The corresponding Hausdorff measure is called the s-dimensional Hausdorff measure and denoted by |$\mathcal{H}^{s}$|⁠. Hausdorff dimension The Hausdorff dimension of a metric space X is denoted and defined by $$ \operatorname{\dim_{\mathsf{H}}} X=\sup\left\{s:\mathcal{H}^{s}(X) > 0\right\}. $$ General references: [10, 18, 24]. Decomposability The following properties will turn crucial. Definition 3.2. Let s > 0. Say that a metric space X is s-decomposable if there is a countable cover |$\left \{X_{n}\right \}$| of X such that |$\operatorname{\dim _{\mathsf{H}}} X_{n} < s$| for all n, and s-indecomposable if it is not s-decomposable. It is easy to show that, for any metric space X, $$\begin{align} \mathcal{H}^{s}(X)>0\Rightarrow X \ \textrm{is} \ s\text{-indecomposable}\Rightarrow\operatorname{\dim_{\mathsf{H}}} X\geqslant s \end{align}$$ (5) and also that none of the implications can be reversed. The first implication, though, has a kind of converse in non-exploding spaces. Note that the non-exploding property is an invariant of nearly Lipschitz equivalence, and, as we shall see below in Lemma 4.2, so is s-indecomposability. Theorem 3.3. If s > 0 and X is a non-exploding analytic metric space, then X is s-indecomposable if and only if it is nearly Lipschitz equivalent to a metric space Z with |$\mathcal{H}^{s}(Z)>0$|⁠. To prove this theorem we prepare two lemmas. Lemma 3.4. Let h be a gauge and |$0<\beta \leqslant \operatorname{ord} h$|⁠. There is a gauge |$\widehat h$| with the following properties: (i) |$\widehat h$| is strictly increasing, (ii) |$\widehat h(r)\geqslant h(r)+r^{\beta }$| on [0, 1], (iii) |$\operatorname{ord}\widehat h=\beta $|⁠, (iv) the gauge |$\widehat h^{1/\beta }$| is subadditive and |$\operatorname{ord}\widehat h^{1/\beta }=1$|⁠, (v) |$\widehat h$| is doubling, and if |$\beta \leqslant 1$|⁠, then |$\widehat h$| is subadditive. Proof. First set h*(r) = h(r) + rβ. It is easy to verify that ord h* = β and clearly |$h^{\ast }\geqslant h$|⁠. Now define $$ \psi(r)= \begin{cases} \displaystyle\sup_{r\leqslant s\leqslant 1}s^{-\beta}h^{\ast}(s), & 0 < r < 1,\\ \displaystyle h^{\ast}(1), & r\geqslant 1. \end{cases} $$ If ψ is bounded, let |$\widehat h(r)=\sup \psi \cdot r^{\beta }$|⁠. In this case everything is trivial. If ψ is unbounded, put |$\widehat h(r)=r^{\beta }\psi (r)$| if > 0 and |$\widehat h(0)=0$|⁠. Routine calculation proves that |$\widehat h$| is right-continuous at 0. First note that obviously |$h^{\ast }\leqslant \widehat h$|⁠. We will often use it. In particular, (ii) holds. (i) Suppose |$0 < r < s \leqslant 1$|⁠. Routine calculation shows that since h* is right-continuous, the supremum in the definition of ψ is attained. Therefore there is |$t\geqslant r$| such that ψ(r) = t−βh*(t). If |$t\geqslant s$|⁠, then ψ(r) = ψ(s) and |$\widehat h(r)<\widehat h(s)$| follow. If t < s, then |$\widehat h(r)=r^{\beta } t^{-\beta }h^{\ast }(t) < h^{\ast }(t)\leqslant h^{\ast }(s)\leqslant \widehat h(s)$|⁠. (iii) |$\operatorname{ord}\widehat h\leqslant \beta $| follows from (ii). We show that |$\operatorname{ord}\widehat h\geqslant \beta $|⁠. Suppose for the contrary that there is ε > 0 such that |$\operatorname{ord}\widehat h<\beta -\varepsilon $|⁠. Since ord h* = β, there is δ > 0 such that h*(s) < sβ−ε for all |$s\leqslant \delta $|⁠. Since ψ is unbounded, there is s0 < δ such that |$s_{0}^{-\beta }h^{\ast }(s_{0})\geqslant \psi (\delta )$|⁠. Since |$\operatorname{ord}\widehat h<\beta -\varepsilon $|⁠, there is r < s0 such that ψ(r) > r−ε. Therefore there is s ∈ [r, 1] such that s−βh*(s) > r−ε and since r < s0, it follows that |$s\leqslant \delta $|⁠. Since for every such s we have h*(s) < sβ−ε, it follows that r−ε < s−βh*(s) < s−βsβ−ε = s−ε. Hence s < r, a contradiction. (iv) Write |$g=\widehat h^{1/\beta }$|⁠. For r < 1 we have |$g(r)=r\sup _{r\leqslant s\leqslant 1}\left (h^{\ast }\right )^{1/\beta }(s)/s$|⁠. Hence g(r)/r is nonincreasing. Therefore $$ g(r+s)=r\tfrac{g(r+s)}{r+s}+s\tfrac{g(r+s)}{r+s}\leqslant r\tfrac{g(r)}{r}+s\tfrac{g(s)}{s}=g(r)+g(s). $$ Since |$\operatorname{ord}\,\widehat h=\beta $|⁠, it is clear that ord g = 1. (v) Both statements can be easily derived from (iv). Lemma 3.5. Let s > 0 and X be a non-exploding analytic space. If X is s-indecomposable, then there is a gauge g with |$\operatorname{ord} g\geqslant s$| and |$\mathcal{H}^{g}(X)>0$|⁠. Proof. Since every ball in a non-exploding space X is obviously totally bounded, its closure in the completion of X is compact. It follows that X is contained in a locally compact, subset of its completion. In particular, it is a subset of a σ-compact space. We may thus employ the following [26, Theorem 6.4] of Sion and Sjerve: if X is an analytic subset of a σ-compact metric space, and |$0 < s_{0} < s_{1} < s_{2} < \dots $| is a sequence of reals, then (1) either there is a cover {Xn} of X such that |$\mathcal{H}^{s_{n}}(X_{n})=0$| for all n, (2) or else there is a gauge g such that |$\mathcal{H}^{g}(X)=\infty $| and ordg > sn for all n. So suppose X is s-indecomposable and pick any sequence sn↗s. Then (1) obviously fails and thus (2) yields a gauge such that |$\mathcal{H}^{g}(X)>0$| and |$\operatorname{ord} g\geqslant \sup s_{n}=s$|⁠. Proof of Theorem 3.3 The forward implication: suppose X is s-indecomposable. By the above lemma there is a gauge g with |$\operatorname{ord} g\geqslant s$| and |$\mathcal{H}^{g}(X)>0$|⁠. Using Lemma 3.4 we may suppose that ord g = s, g is strictly increasing and |$g(r)\geqslant r^{s}$|⁠. Define a gauge h(r) = g(r)1/s. By Lemma 3.4 we may also suppose that h is strictly increasing, subadditive, |$h(r)\geqslant r$|⁠, and ord h = 1. Let d be the metric of X. Define a new metric on X by ρ(x, y) = h(d(x, y)). It is indeed a metric inducing the same topology, because h is subadditive and strictly increasing. The identity mapping (X, ρ) → (X, d) is Lipschitz, because |$h(r)\geqslant r$|⁠. The identity mapping (X, d) → (X, ρ) is nearly Lipschitz, because ord h = 1. Thus (X, d) is nearly Lipschitz equivalent to (X, ρ). Since ρs(x, y) = g(d(x, y)), we have |$\mathcal{H}^{s}(X,\rho )=\mathcal{H}^{g}(X,d)>0$|⁠. The reverse implication is easy: let Z be nearly Lipschitz equivalent to X and |$\mathcal{H}^{s}(Z)>0$|⁠. Then Z is s-indecomposable by (5) and, by Lemma 4.2 below, so is X. 4 Mapping Non-exploding Spaces onto Self-similar Sets and Cubes We are ready to present and prove the second summit of the paper: an analytic non-exploding space maps onto the cube [0, 1]n by a nearly Lipschitz map if and only if it is n-indecomposable. We will actually prove a bit more general statement that involves self-similar sets. The material is taken from [28], where a self-similar set is defined as the attractor of an iterated function system with all functions being contracting similarities of |$\mathbb{R}^{n}$|⁠. We also a priori impose upon self-similar sets the Open Set Condition. All of the relevant definitions can be found in [28]. Interested readers are referred to one of the books [10, 11, 18] for an overview of self-similar sets and related material. Definition 4.1. A mapping between metric spaces f : X → Y is termed dimension preserving if |$\operatorname{\dim _{\mathsf{H}}} f(E)\leqslant \operatorname{\dim _{\mathsf{H}}} E$| for every set E ⊆ X. We do not a priori impose any continuity on the mapping. It is well-known and easy to see that Lipschitz mappings are dimension preserving (see, e.g., [10, Lemma 6.1]) and it is also easy to see that nearly Lipschitz mappings are also dimension preserving. It is also worth noticing that dimension-preserving maps preserve s-decomposability. Proof is straightforward. Lemma 4.2. Let f : X → Y be a dimension preserving mapping onto Y. If X is s-decomposable, then so is Y. In particular, the conclusion holds if f is nearly Lipschitz. Theorem 4.3. Let X be a non-exploding analytic metric space and |$S\subseteq \mathbb{R}^{n}$| a self-similar set satisfying the Open Set Condition. Let |$s=\operatorname{\dim _{\mathsf{H}}} S$|⁠. The following are equivalent: (i) X is s-indecomposable. (ii) There is a nearly Lipschitz mapping |$f:X\to \mathbb{R}^{n}$| such that S ⊆ f[X]. (iii) There is a dimension preserving surjection g : X → S. Proof. (ii)⇒(iii) is easy: start with the nearly Lipschitz map f. Pick a single point z ∈ S and define g(x) = f(x) if x ∈ f−1(S) and g(x) = z otherwise. Since, as pointed out above, f is dimension preserving, so is g. (iii)⇒(i) is also easy: suppose that (iii) holds and yet X is s-decomposable. By Lemma 4.2, S is s-decomposable as well. But self-similar sets with the Open Set Condition are indecomposable because they have positive Hausdorff measure, see,e.g., [10]. The only remaining implication (i)⇒(ii) is harder. We postpone its proof until we gather some background material. Corollary 4.4. Let X be a non-exploding analytic metric space and n ∈ ω. The following are equivalent: (i) X is n-indecomposable. (ii) There is a nearly Lipschitz surjection f : X → [0, 1]n. (iii) There is a dimension preserving surjection g : X → [0, 1]n. Proof. This is immediate, because the cube [0, 1]n is a self-similar set with the Open Set Condition. We only need to take care of values of f that are outside [0, 1]n. But since [0, 1]n is convex, we may send every such point to the closest point on the boundary of [0, 1]n. This mapping is Lipschitz, which is enough, since a composition of a nearly Lipschitz map and a Lipschitz map is nearly Lipschitz. Let us point out that the above theorem and corollary apply in particular to Borel sets in Euclidean spaces. We present an illustration: Corollary 4.5. Let |$m\leqslant n$| be positive integers. For every m-indecomposable Borel set |$X\subseteq \mathbb{R}^{n}$| there is a nearly Lipschitz surjection f : X → [0, 1]m. In particular, such a mapping exists whenever |$\mathcal{H}^{m}(X)>0$|⁠. So in particular, the Vitushkin example mentioned in the introduction (i.e., a compact set |$X\subseteq \mathbb{R}^{2}$| with positive linear measure that cannot be mapped onto a segment by a Lipschitz map) maps onto [0, 1] by a nearly Lipschitz mapping. We are aiming toward the proof of the remaining part of Theorem 4.3. We prepare a couple of notions and lemmas. Monotone spaces At this point we make use of the notion of monotone metric space introduced in [28], developed in [23] and further investigated in a number of papers, e.g., [13, 14, 20–22]. By the definition, a metric space (X, d) is monotone if there is a linear order < on X and a constant c > 0 such that x < y < z ⇒ d(x, y) < c d(x, z). The following two facts are crucial for our proof. Lemma 4.6. ([28, Theorem 4.5, Lemma 3.2]) Let X be an analytic monotone metric space and |$S\subseteq \mathbb{R}^{m}$| a self-similar set. Let |$s=\operatorname{\dim _{\mathsf{H}}} S$|⁠. If |$\mathcal{H}^{s}(X)>0$|⁠, then there is compact set K ⊆ X such that |$\mathcal{H}^{s}(K)>0$| and a nearly Lipschitz mapping g : K → S onto S. Lemma 4.7. ([16, 23]) Every ultrametric space is monotone. We will also need to extend nearly Lipschitz mappings. We prove the extension lemma in a slightly more general setting. Let us call a mapping f between metric spaces nearly β-Hölder if it is α-Hölder for all α < β. This clearly extends the notion of nearly Lipschitz mapping. Proposition 2.3 has a counterpart for nearly Hölder mappings: Lemma 4.8. A mapping f : (X, dX) → (Y, dY) is nearly β-Hölder if and only if there is a Hausdorff function h such that |$\operatorname{ord} h\geqslant \beta $| and $$ d_{Y}(\,f(x),f(\,y))\leqslant h(d_{X}(x,y))\quad \textrm{for all} \ x,y\in X. $$ It is no surprise that nearly Lipschitz and nearly Hölder mappings are extendable: Lemma 4.9. Let X be a metric space and Y ⊆ X. Let β ≤ 1. Any nearly β-Hölder mapping |$f:Y\to \mathbb{R}^{m}$| extends to a nearly β-Hölder mapping over X. Proof. By Lemma 4.8 and 3.4 there is a subadditive Hausdorff function h such that ordh = β and |$\left \lvert\, f(x)-f(y)\right \rvert \leqslant h(d(x,y))$| for all x, y ∈ Y. It is obviously enough to prove the statement for the coordinates of f, so assume without loss of generality that |$f:Y\to \mathbb{R}$|⁠. Define the extension |$f^{\ast }:X\to \mathbb{R}$| by $$ f^{\ast}(x)=\inf_{z\in Y} f(z)+h\left(d(x,z)\right). $$ Proving that f*(x) = f(x) for all x ∈ X is straightforward. We prove that f* is nearly β-Hölder. Let x, y ∈ X. Since h is subadditive, we have $$ f(z)+h(d(x,z))\leqslant f(z)+h(d(y,z))+h(d(x,y))\quad \textrm{for all}\ z\in Y. $$ Therefore |$f^{\ast }(x)\leqslant f^{\ast }(\,y)+h(d(x,y))$| and thus $$ \left\lvert\, f^{\ast}(x)-f^{\ast}(\,y)\right\rvert\leqslant h(d(x,y)) \quad \textrm{for all} \ x,y\in X. $$ Apply Lemma 4.8 to conclude that f* is nearly β-Hölder. Proof of Theorem 4.3 (i)⇒(ii). Suppose X is s-indecomposable. By Theorem 3.3, we may suppose that |$\mathcal{H}^{s}(X)>0$|⁠. By the Howroyd theorem (Theorem 3.1) there is a finite Borel measure |$\mu \leqslant \mathcal{H}^{s}$| such that μ(X) > 0. Apply Theorem 2.6: there is a nearly ultrametric compact set N ⊆ X such that |$\mathcal{H}^{s}(N)\geqslant \mu (N)>0$|⁠. By Proposition 2.5, there exists an ultrametric space U and a nearly Lipschitz surjection ϕ : N → U, with a Lipschitz inverse. Since U is a Lipschitz preimage of N, we have |$\mathcal{H}^{s}(U)>0$|⁠. By Lemma 4.7, U is monotone, therefore Lemma 4.6 yields a compact subset C ⊆ U and a nearly Lipschitz mapping g : C → S onto S. The composed map g ∘ ϕ : N → S onto S is clearly nearly Lipschitz. Now it is enough to apply Lemma 4.9 to extend g ∘ ϕ over X. 5 Comments and Questions Nearly Hölder mappings We present a mild generalization of Theorem 4.3. The dimension preserving property can be parameterized as follows: for a mapping f : (X, dX) → (Y, dY) between metric spaces define its Hausdorff dimension by $$ \operatorname{\dim_{\mathsf{H}}}\ f=\inf\left\{\alpha:\operatorname{\dim_{\mathsf{H}}} f(E)\leqslant\alpha\operatorname{\dim_{\mathsf{H}}} E \ \textrm{for every} \ E\subseteq X\right\}. $$ It is routine to show that if f is nearly s-Hölder, then |$\operatorname{\dim _{\mathsf{H}}}\ f\leq 1/s$|⁠. Theorem 5.1. Let X be a non-exploding analytic metric space and |$S\subseteq \mathbb{R}^{m}$| a self-similar set; let |$s=\operatorname{\dim _{\mathsf{H}}} S\geqslant t>0$|⁠. The following are equivalent: (i) X is t-indecomposable. (ii) There is a nearly |$\frac ts$|-Hölder mapping |$f:X\to \mathbb{R}^{m}$| such that S ⊆ f[X]. (iii) There is a surjection f : X → S such that |$\operatorname{\dim _{\mathsf{H}}} f\leqslant \frac st$|⁠. Proof in outline. (ii)⇒(iii) is obvious and (iii)⇒(i) follows easily by modification of Lemma 4.2. (i)⇒(ii) follows the same way as in the proof of Theorem 4.3, one only has to insert between the spaces U and S an ultrametric space |$U^{\prime }=\left (U,d^{s/t}\right )$| where d is the ultrametric of U. Corollary 5.2. Let X be a non-exploding analytic metric space. Let |$t\leqslant m$|⁠. The following are equivalent: (i) X is t-indecomposable. (ii) There is a nearly |$\frac tm$|-Hölder surjection f : X → [0, 1]m. (iii) There is a surjection f : X → [0, 1]m such that |$\operatorname{\dim _{\mathsf{H}}} f\leqslant \frac mt$|⁠. Corollary 5.3. (Peano curves) For any m > n > 0 there is a nearly |$\frac nm$|-Hölder Peano curve, i.e., a surjection p : [0, 1]n → [0, 1]m. This seems to have attracted some attention. E.g., Arnold [3, Problem 1988-5] claims that without a proof and asks if “nearly” can be dropped. Semmes [25, 9.1] also discusses this topic. The condition |$s\geqslant t$| in the above theorem is not necessary, it is only needed for the extension of the nearly |$\frac ts$|-Hölder mapping over the whole space X, guaranteed by Lemma 4.9 for nearly β-Hölder mappings only when β ≤ 1. Inspection of the proofs shows that the following remains true for any t > 0. Recall that we a priori impose upon self-similar sets the Open Set Condition. Theorem 5.4. Let X be a non-exploding analytic metric space and |$S\subseteq \mathbb{R}^{m}$| a self-similar set; let |$s=\operatorname{\dim _{\mathsf{H}}} S$| and t > 0. The following are equivalent: (i) X is t-indecomposable. (ii) There is a compact set K ⊆ X and a nearly |$\frac ts$|-Hölder surjection f : K → S. (iii) There is a surjection f : X → S such that |$\operatorname{\dim _{\mathsf{H}}} f\leqslant \frac st$|⁠. Lipschitz and Hölder maps As mentioned in the introduction, in [16] it was proved that any analytic metric space X with |$\operatorname{\dim _{\mathsf{H}}} X>n$| maps onto the cube [0, 1]n by a Lipschitz map. The proof builds upon ideas similar to those in the present paper, but deviates in one detail: the constructed mapping factorizes through an interval. Thus it is inevitably short when mappings onto disconnected self-similar sets are under consideration. However, there is, as indicated below, an easy remedy. Theorem 5.5. Let X be an analytic metric space and |$S\subseteq \mathbb{R}^{n}$| a self-similar set. If |$\operatorname{\dim _{\mathsf{H}}} X>\operatorname{\dim _{\mathsf{H}}} S$|⁠, then there is a compact set C ⊆ X and a Lipschitz surjection f : C → S. Proof. By the theorem of Mendel and Naor [19] quoted above there is a compact set C ⊆ X, a compact ultrametric space U and a Lipschitz bijection g : C → U such that |$\operatorname{\dim _{\mathsf{H}}} C=\operatorname{\dim _{\mathsf{H}}} U>\operatorname{\dim _{\mathsf{H}}} S$|⁠. By Lemma 4.7 and [28, Theorem 4.7 and Lemma 3.2], there is a Lipschitz surjection ϕ : U → S. The required mapping is of course f = ϕ ∘ g. Dimension preserving versus nearly Lipschitz As to nearly Lipschitz and dimension-preserving mappings, we do not really know anything of the relation of the notions except of the following elementary fact. Proposition 5.6. A nearly Lipschitz mapping is dimension preserving and continuous. However, the theorems we proved indicate that some converse to this proposition might hold. For example, Corollary 4.4 contains the following information: let X be an analytic, non-exploding space. If there is a dimension preserving surjection f : X → [0, 1]m, then there is a nearly Lipschitz surjection f : X → [0, 1]m. It is thus natural to ask: Question 5.7. Is there any, however partial, converse to Proposition 5.6? Non-exploding spaces The notion of non-exploding space is tailored so that the proof of Theorem 2.6 works. The paramount question is, of course, whether this condition is really needed. Question 5.8. Is it true that for every analytic (or, equivalently, compact) metric space X and every finite Borel measure μ on X there is a nearly ultrametric set C ⊆ X such that μ(C) > 0? Note that if the answer were affirmative, the theorem of Balka, Darji, and Elekes [6] quoted in the introduction (and its consequences) would hold for all compact metric spaces. Even a little improvement would be valuable. Maybe there is some room for improvement. First, there may be a more effective way of constructing a bi-Lipschitz embedding of a compact space into |$\ell ^{\infty }$|⁠. And maybe there is another less regular construction of a nearly ultrametric set in |$\mathbb{T}$| with positive Haar measure that does not require the diameters of circles to tend to zero according to a slow function G. The importance of the following question is that all analytic spaces with finite box-counting dimension satisfy the condition and thus Theorem 2.6 would hold for all analytic spaces with finite packing dimension (since they are countable unions of sets with finite box-counting dimension). Question 5.9. Let X be an analytic metric space. Suppose it satisfies condition $$ \lim_{r\to0}\frac{\log Q(r)}{\log r}<\infty $$ in place of condition (NE). Is it true that for every finite Borel measure μ on X there is a nearly ultrametric set C ⊆ X such that μ(C) > 0? Let us note that an affirmative answer to either of the above Questions 5.8 and 5.9 would extend the results of [5]. The other main result, Theorem 4.3, depends mostly on the existence of large nearly ultrametric sets. That is another reason to study Questions 5.8 and 5.9. Positive Hausdorff measure We know from [16] that for every analytic metric space X, if |$\operatorname{\dim _{\mathsf{H}}} X>m$|⁠, then there is a Lipschitz surjection of X onto [0, 1]m, and that a non-exploding analytic X is s-indecomposable if and only if there is a nearly Lipschitz surjection of X onto [0, 1]m. The condition |$\mathcal{H}^{m}(X)>0$| is between the two. Question 5.10. Is there a condition similar to m-indecomposability that characterizes (non-exploding) compact spaces that map onto [0, 1]m by a Lipschitz mapping? Question 5.11. Is there a type of mapping such that the existence of such a mapping of X onto [0, 1]m or some similar condition characterizes (non-exploding) compact spaces with |$\mathcal{H}^{m}(X)>0$|? Funding This work was supported by the Grant Agency of the Czech Republic [GAČR 15–082185]. Acknowledgments I would like to thank Richárd Balka, Márton Elekes, and Tamás Keleti for inspiring discussions while I was visiting Institute of Mathematics of Eötvös Loránd University. In particular, I am grateful to Richárd Balka who pointed me to the paper [26] and to Richárd Balka and Márton Elekes for making me finally write the paper. I am also grateful to the two anonymous referees for their valuable comments. Finally, I would like to thank the College of Wooster. The paper was finalized during my visit thereby. In particular, my special thanks go to Pam Pierce. References [1] Aharoni , I. “Every separable metric space is Lipschitz equivalent to a subset of |$c_0^+$|⁠.” Isr. J. Math. 19 ( 1974 ): 284 -- 91 . MR 0511661 . Google Scholar Crossref Search ADS WorldCat [2] Alberti , G. , M. Csörnyei , and D. Preiss . “Structure of null sets in the plane and applications.” Eur. Math. Soc. Zürich : European Congress of Mathematics ( 2005 ): 3 -- 22 . MR 2185733 (2007c:26014) . OpenURL Placeholder Text WorldCat [3] Arnold , V. I. Arnold’s Problems . Translated and revised edition of the 2000 Russian original, with a preface by V. Philippov, A. Yakivchik, and M. Peters. Berlin : Springer , 2004 , MR 2078115 . [4] Assouad , P. “Remarques sur un article de Israel Aharoni sur les prolongements lipschitziens dans c0 (Israel J. Math. 19 (1974), 284–291).” Isr. J. Math. 31 , no. 1 ( 1978 ): 97 -- 100 . MR 0511662 . [5] Assouad , P. “Plongements lipschitziens dans Rn.” Bull. Soc. Math. France. 111 , no. 4 ( 1983 ): 429 -- 48 . MR 763553 . [6] Balka , R. , U. B. Darji , and M. Elekes . “Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps.” Adv. Math. 293 ( 2016 ): 221 -- 74 . Google Scholar Crossref Search ADS WorldCat [7] Besicovitch , A. S. “On existence of subsets of finite measure of sets of infinite measure.” Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math . 14 ( 1952 ): 339 -- 44 . MR 0048540 (14,28e) . [8] Christensen , J. P. R. “On sets of Haar measure zero in abelian Polish groups.” Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972) , vol. 13, 255 -- 60 . 1972 . MR 0326293 . [9] Davies , R. O. “Subsets of finite measure in analytic sets.” Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math . 14 ( 1952 ): 488 -- 89 . MR 0053184 (14,733g) . [10] Falconer , K. J. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics , vol. 85 . Cambridge : Cambridge University Press , 1986 . MR 867284 (88d:28001) . [11] Falconer , K. J. Fractal Geometry: Mathematical Foundations and Applications , 2nd ed. Hoboken, New Jersey: John Wiley & Sons Inc. MR 2118797 (2006b:28001) . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC [12] Howroyd , J. D. “On dimension and on the existence of sets of finite positive Hausdorff measure.” Proc. London Math. Soc. 3. 70 , no. 3 ( 1995 ): 581 -- 604 . MR 1317515 (96b:28004) . [13] Hrusák , M. , T. Mátrai , A. Nekvinda , V. Vlasák , and O. Zindulka . “Properties of functions with monotone graphs.” Acta Math. Hungar. 142 , no. 1 ( 2014 ): 1 -- 30 . MR 3158849 . Google Scholar Crossref Search ADS WorldCat [14] Hrusák , M. and O. Zindulka . “Cardinal invariants of monotone and porous sets.” J. Symbolic Logic. 77 , no. 1 ( 2012 ): 159 -- 73 . MR 2951635 . [15] Keleti , T. “A peculiar set in the plane constructed by Vituškin, Ivanov and Melnikov.” Real Anal. Exchange. 20 , no. 1 ( 1994/95 ): 291 -- 312 . MR 1313694 . [16] Keleti , T. , A. Máthé , and O. Zindulka . “Hausdorff dimension of metric spaces and Lipschitz maps onto cubes.” Int. Math. Res. Not. no. 2 ( 2014 ): 289 -- 302 . MR 3159074 . [17] Laczkovich , M. “Paradoxical decompositions using Lipschitz functions.” Real Anal. Exchange. 17 , no. 1 ( 1991/92 ): 439 -- 44 . MR 1147388 (93f:28008) . [18] Mattila , P. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability Cambridge Studies in Advanced Mathematics , vol. 44 . Cambridge : Cambridge University Press , 1995 . MR 1333890 (96h:28006) . [19] Mendel , M. and A. Naor . “Ultrametric subsets with large Hausdorff dimension.” Invent. Math. 192 , no. 1 ( 2013 ): 1 -- 54 . MR 3032324 . [20] Nekvinda , A. “A Cantor set in the plane and its monotone subsets.” Acta Math. Hungar. 148 , no. 1 ( 2016 ): 43 -- 55 . MR 3439281 . Google Scholar Crossref Search ADS WorldCat [21] Nekvinda , A. , D. Pokorný , and V. Vlasák . “Some results on monotone metric spaces.” J. Math. Anal. Appl. 413 , no. 2 ( 2014 ): 999 -- 1016 . MR 3159817 . Google Scholar Crossref Search ADS WorldCat [22] Nekvinda , A. and O. Zindulka . “A Cantor set in the plane that is not sigma-monotone.” Fund. Math. 213 , no. 3 ( 2011 ): 221 -- 32 . MR 2822419 . [23] Nekvinda , A. and O. Zindulka . “Monotone metric spaces.” Order. 29 , no. 3 ( 2012 ): 545 -- 58 . MR 2979649 . [24] Rogers , C. A. Hausdorff Measures. London-New York : Cambridge University Press , 1970 . MR 0281862 . [25] Semmes , S. “Where the buffalo roam: infinite processes and infinite complexity.” arXiv:math/0302308v1(2003) . [26] Sion , M. and D. Sjerve . “Approximation properties of measures generated by continuous set functions.” Mathematika. 9 ( 1962 ): 145 -- 56 . MR 0146331 . Google Scholar Crossref Search ADS WorldCat [27] Vitushkin , A. G. , L. D. Ivanov , and M. S. Melnikov . “Incommensurability of the minimal linear measure with the length of a set.” Dokl. Akad. Nauk SSSR. 151 ( 1963 ): 1256 -- 59 . MR 0154965 (27 4908) . [28] Zindulka , O. “Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps.” Fund. Math. 218 , no. 2 ( 2012 ): 95 -- 119 . MR 2957686 . Google Scholar Crossref Search ADS WorldCat © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Mapping Borel Sets onto Balls and Self-similar Sets by Lipschitz and Nearly Lipschitz Maps
JO - International Mathematics Research Notices
DO - 10.1093/imrn/rny008
DA - 2020-02-07
UR - https://www.deepdyve.com/lp/oxford-university-press/mapping-borel-sets-onto-balls-and-self-similar-sets-by-lipschitz-and-sZ0NDPXw46
SP - 698
VL - 2020
IS - 3
DP - DeepDyve
ER -