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Geometric progression stabilizer in a general metric S. A. Bogatyi Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 3, DOI: https://doi.org/10.4213/sm9782 
Bibliographic databases:
    § 1. IntroductionThe paper is concerned with the geometry of the Gromov-Hausdorff distance (see [1]–[3]) on the class of all nonempty metric spaces. Given a metric space $(X,\varrho )$ and $r>0$, the open $r$-neighbourhood of a subset $A$ of $ X$ is by definition $U_r(A)=\bigl\{x\in X\colon \varrho (x,A)<r\bigr\}$. The Hausdorff distance between subsets $A$ and $B$ of $ X$ is
$$
\begin{equation*}
d_H(A,B)=\inf\bigl\{r\colon A\subset U_r(B),\,B\subset U_r(A)\bigr\}.
\end{equation*}
\notag
$$
The Hausdorff distance is a generalized pseudometric, that is, it can be infinite and can also be zero for distinct subsets. Given two metric spaces $X$ and $Y$, the Gromov-Hausdorff distance is defined by
$$
\begin{equation*}
d_{\mathrm{GH}}(X,Y)=\inf\bigl\{d_H(X',Y')\colon X',Y'\subset Z,\, X'\approx X,\,Y'\approx Y\bigr\},
\end{equation*}
\notag
$$
where the notation $X\approx X'$ for metric spaces $X$ and $X'$ means that they are isometric. Theorem 1.1. The Gromov-Hausdorff distance is a generalized pseudometric which vanishes on pairs of isometric spaces. Let $X$ and $Y$ be arbitrary sets. A multivalued map $R\colon X\to Y$ is uniquely determined by its graph, for which we also use the notation $R$: Clearly, the graphs of multivalued maps are precisely subsets $R\subset X\times Y$ such that for each point $x\in X$ there exists $y\in Y$ such that $(x,y)\in R$. We also call such sets $R\subset X\times Y$ complete relations. To simplify the notation, for points in $R(x)$ we also use the notation $y_x$. In metric geometry surjective multivalued maps are called correspondences. Given a correspondence $R$, the graph of the inverse map $R^{-1}$ is a subset of $Y\times X$, so we denote it by $R^*$. The set of correspondences between $X$ and $Y$ is denoted by $\mathcal R(X,Y)$.Given a complete relation $R\subset X\times Y$ of metric spaces $(X,\varrho_X)$ and $(Y,\varrho_Y)$, its distortion is
$$
\begin{equation}
\operatorname{dis} R=\sup\bigl\{|\varrho_X(x,x')-\varrho_Y(y,y')|\colon (x,y),\,(x',y')\in R\bigr\}.
\end{equation}
\tag{1}
$$
It is convenient to estimate the Gromov-Hausdorff distance in terms of the distortions of correspondences (see [3]). Recall (see [4] and [5]) that by definition the cloud $[X]$ of a metric space is the class
$$
\begin{equation*}
[X]=\bigl\{Y\colon d_{\mathrm{GH}}(X,Y)<\infty \bigr\}.
\end{equation*}
\notag
$$
For a metric space $(X,\varrho )$ and a positive number $\lambda$ we denote by $\lambda X$ the similar space $(X,\lambda \varrho )$, which is the set $X$ all distances in which are multiplied by $\lambda $. The group of similarities $(\mathbb R_+,\times )$ acts on the class of metric spaces considered up to isometries, up to zero distances and up to finite distances. It was shown in [4] and [5] that $\lambda X$ can lie on an infinite Gromov-Hausdorff distance from the original space $X$. Thus, given a metric space $X$, three different stabilizers can be defined. We set
$$
\begin{equation*}
\begin{gathered} \, \operatorname{St}X=\{\lambda\in\mathbb R_+\colon \lambda X\ \text{is isometric to }X\}, \\ \operatorname{St}_0X=\bigl\{\lambda\in\mathbb R_+\colon d_{\mathrm{GH}}(X,\lambda X)=0\bigr\} \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\operatorname{St}[X]=\bigl\{\lambda\in\mathbb R_+\colon d_{\mathrm{GH}}(X,\lambda X)<\infty\bigr\}.
\end{equation*}
\notag
$$
Finally, similarity transformations are defined on a normed vector space $V$. We denote the corresponding stabilizer of a subset $X$ of $V$ (which is an element of $2^V$) by $\operatorname{St}_V X$. Thus, $\operatorname{St}_V X=\{\lambda\in\mathbb R_+\colon \lambda X=X\}$. It is obvious that all four stabilizers are subgroups and
$$
\begin{equation*}
\operatorname{St}_VX\subseteq \operatorname{St}X\subseteq \operatorname{St}_0X\subseteq \operatorname{St}[X]\subseteq (\mathbb R_+,\times ).
\end{equation*}
\notag
$$
It is also important to note that
$$
\begin{equation*}
\begin{gathered} \, \text{if two metric spaces }X\text{ and }Y\text{ are isometric, then }\operatorname{St}X=\operatorname{St}Y; \\ \text{if }d_{\mathrm{GH}}(X,Y)=0,\quad\text{then }\operatorname{St}_0X=\operatorname{St}_0Y; \\ \text{if }d_{\mathrm{GH}}(X,Y)<\infty,\quad\text{then }\operatorname{St}[X]=\operatorname{St}[Y]. \end{gathered}
\end{equation*}
\notag
$$
If $\operatorname{St}[X]\ne \{1\}$, then the centre of the cloud
$$
\begin{equation*}
Z[X]=\bigl\{Y\in [X]\colon (Y \text{ is complete and})\ \operatorname{St}_0Y=\operatorname{St}[X]\bigr\}
\end{equation*}
\notag
$$
is well defined (see [5], Corollary 12). If $\operatorname{St}[X]\ne \{1\}$, then $Z[X]\ne \varnothing $ and $d_{\mathrm{GH}}(Y_1,Y_2)=0$ for any $Y_1,Y_2\in Z[X]$. Conversely, if $d_{\mathrm{GH}}(Y_1,Y_2)=0$ and $Y_1\in Z[X]$, then $Y_2\in Z[X]$. We can describe the centre of a cloud as the family of all spaces in the cloud which have a nontrivial stabilizer $\operatorname{St}_0Y$. For such a space $Y$ its stabilizer $ \operatorname{St}_0Y$ is maximal possible among the spaces in this cloud: it is equal to $\operatorname{St}[Y]=\operatorname{St}[X]$. For each $\lambda \in \operatorname{St}[X]$ we can produce a space $X_\lambda $ such that $X_\lambda $ is isometric to $\lambda X_\lambda $ but for different $\lambda,\mu \in \operatorname{St}[X]$ the spaces $X_\lambda $ and $X_\mu $ need not be isometric. Moreover, the geometric stabilizer $\operatorname{St}Y$ of any space $Y\in [(\mathbb Q_+,\varrho_1)]$ in the cloud of the hedgehog of positive rational numbers (the set of positive rational numbers with metric (4), where $\alpha=1$) is at most countable (see [5], Corollary 26), so it is strictly smaller than the stabilizer $\operatorname{St}[Y]=\operatorname{St}[(\mathbb Q_+,\varrho_1)]=\mathbb R_+$ of this cloud. We assume in what follows that $X,Y \subset [0,\infty )$, $0 \in X$ and $0 \in Y$. We denote the point $0$ in the set $X$ by $0_X$. Since we consider various metrics on these sets, we will also use the notation $\operatorname{dis}_{\varrho_X,\varrho_Y}R$ where necessary. Traditionally, in the geometry of the Gromov-Hausdorff metric the distance $\varrho(x,x')$ between two points is denoted by $|xx'|$. We are interested in normalized metrics, that is, metrics $\varrho $ on $X$ such that We denote such a space by $X_{\varrho}$, and we denote the set of normalized metrics on $X$ by $\mathcal M_X$, or just by $\mathcal M$ when $X$ is fixed.It follows from the triangle inequality that for any points $x,x'\geqslant 0$ we have
$$
\begin{equation}
x-x'=\varrho(x,0_X)-\varrho(x',0_X)\leqslant \varrho(x,x')\leqslant \varrho(x,0_X)+\varrho(0_X,x')=x+x'.
\end{equation}
\tag{3}
$$
In both extremal cases we have interesting examples. Left-hand equality (for all $x>x'$) corresponds to the metric on $X$ inherited from the real line. Right-hand equality (for all $x\ne x'$) corresponds to the discrete hedgehog $\widehat{X}$ (see [5]). Intermediate ‘linear’ metrics also produce important examples. For each $\alpha$, $-1\leqslant \alpha \leqslant 1$, consider the following metric on the set of nonnegative numbers (and thus on each set $X$ under consideration):
$$
\begin{equation}
\varrho_\alpha (x,x')=x+\alpha x'=\frac{1-\alpha}{2}|x-x'|+\frac{1+\alpha}{2}(x+x') \quad\text{for } x'<x.
\end{equation}
\tag{4}
$$
Clearly, (4) defines a metric on the set of nonnegative real numbers. Inequalities (3) can be stated as the assertion that for each metric $\varrho \in \mathcal M_X$ we have ${\varrho_{-1} \leqslant \varrho \leqslant \varrho_1}$. The metric $\varrho_{0}$ is also distinguished by certain special properties. Recall that a metric $\varrho $ is called an ultrametric or a non-Archimedean metric if
$$
\begin{equation*}
\varrho (x,y)\leqslant \max\{\varrho (x,z),\varrho (z,y)\} \quad\text{for any three points } x,y,z.
\end{equation*}
\notag
$$
Proof. Let $\varrho $ be a normalized ultrametric in $X$. Then In a similar way Since $x'<x$, we have $x\leqslant \max\{\varrho (x,x'),x'\}=\varrho (x,x')$.
The proof is complete. It is obvious that for $\lambda \ne 1$ and $\varrho \in \mathcal{M}_X$ the metric $\lambda \varrho $ is not normalized any longer. For metric spaces $X\subset [0,\infty )$ we consider it natural to identify the metric $\lambda \varrho_X$ on $X$ with the metric $\varrho_{Y}(y,y')=\lambda \varrho_X(\lambda^{-1}y,\lambda^{-1}y')$ on $Y=\lambda X$, where $\lambda X$ denotes the result of multiplication of $X\subset [0,\infty )$ by $\lambda $. That is, instead of introducing the new metric $\lambda \varrho $ on $X$ it is convenient to mean by $\lambda X$ the metric on the new set $\lambda X$. Proposition 1.4. The map $\lambda \colon X\to Y$ is an isometry between the metric space $\lambda X=(X,\lambda \varrho_{X})$ and $Y=\bigl(Y=\lambda X,\varrho_Y(y,y')=\lambda\varrho_X(\lambda^{-1}y,\lambda^{-1}y')\bigr)$, and $\varrho_Y\in \mathcal{M}_Y$. Proof. Since it follows that $\varrho_{Y}\in \mathcal{M}_{Y}$. Because the map $\lambda\colon X\to Y$ is an isometry of $(X,\lambda \varrho_X)$ onto $(Y,\varrho_Y)$.
The proof is complete. Using Proposition 1.4 we can reduce the similarity of metrics to the similarity of subsets of the real line with the help of Theorem 2.1. Each normalized metric $\varrho_X$ defines a function $\alpha \colon X\times X\to \mathbb R$ such that $\varrho_X(x,x')=\max\{x,x'\}+\alpha (x,x')\min\{x,x'\}$. If the context is unambiguous, then it will be denoted by $\varrho_\alpha $. The functions $\alpha (x,x')$ provide a convenient parametrization for $\mathcal M_X$. All these functions lie (by Proposition 3.5) in a closed ball of radius $1$ with respect to the $\sup $-norm, so even when some metrics in $\mathcal M_X$ lie at an infinite Gromov-Hausdorff distance from each other we can discuss the topology and metric on $\mathcal M_X$. The following result is certainly of interest but we do not use it below and so present no proof. Proposition 1.5. The set of normalized metrics $\mathcal M$ is convex and closed in the linear space of maps $\operatorname{Map}(X\times X,\mathbb R)$. It is closed in the topology of uniform convergence and the topology of pointwise convergence on $ \operatorname{Map}(X\times X,\mathbb R)$ alike. The case of a geometric progression is of particular interest for us. Closed subgroups of the group $(\mathbb R_+,\times )$ of similarities are stabilizers of the geometric progression, on which, apart from the linear metrics (4), it is reasonable to consider also more general invariant metrics. For $p>1$ let
$$
\begin{equation*}
G_p=\bigl\{p^n\colon n\in \mathbb Z\bigr\}\subset (\mathbb R_+,\times )
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
X_p=\overline {G_p}=\{0\}\cup G_p=\bigl\{p^n\colon n\in \{-\infty \}\cup \mathbb Z\bigr\}\subset [0,\infty ).
\end{equation*}
\notag
$$
We denote the set $\mathcal M_{X_p}$ by $\mathcal M_p$. In Gromov-Hausdorff geometry invariant normalized metrics play an important role. We call $\varrho \in \mathcal M_p$ an invariant normalized metric if
$$
\begin{equation*}
\varrho (p^m,p^n)=p\varrho (p^{m-1},p^{n-1}) \quad\text{for all } n,m\in \{-\infty \}\cup \mathbb Z.
\end{equation*}
\notag
$$
We denote the set of invariant normalized metrics on $X_p$ by $\mathcal{IM}_p$. All metrics $\varrho_\alpha $, $-1\leqslant \alpha \leqslant 1$, are obviously invariant. Proposition 1.6. The set of invariant normalized metrics $\mathcal{IM}_p$ is convex and closed in the space of normalized metrics $\mathcal M_p$. It is closed in the topology of uniform convergence and the topology of pointwise convergence on $\mathcal M_p$ alike. § 2. Main resultsThe following theorem shows that, apart from being minimal, the metric $\varrho_{-1}$ produces the ‘maximal’ cloud from the standpoint of ‘carriers’. Theorem 2.1. If $\varrho_{X}\in \mathcal{M}_X$, $\varrho_{Y}\in \mathcal{M}_Y$ and $d_{\mathrm{GH}}((X,\varrho_{X}),(Y,\varrho_{Y}))<\infty $, then Complete metric spaces lying at Gromov-Hausdorff distance zero from each other are not necessarily isometric nor even homeomorphic. The following theorem shows that the set $\mathcal{M}_p$ is ‘closed’ in the Gromov-Hausdorff class. Theorem 2.2. For a metric $\varrho \!\in\! \mathcal{M}_p$ and a complete metric space $Z$, if $d_{\mathrm{GH}}((X_p,\varrho), (Z,\varrho_{Z}))=0$, then the spaces $(X_p,\varrho )$ and $(Z,\varrho_{Z})$ are isometric. Corollary 2.3. For a metric $\varrho \in \mathcal{M}_p$ and an incomplete metric space $Z$, if $d_{\mathrm{GH}}((X_p,\varrho ),(Z,\varrho_{Z}))=0$, then $(Z,\varrho_{Z})$ is isometric to the space $(X_p,\varrho )$ without the point $0$. It obviously follows from Theorem 2.2 and Corollary 2.3 that the centre of the cloud of a geometric progression with nontrivial stabilizer consists of two spaces, a complete and an incomplete one. Namely, it contains the completion of the progression $X_p=\overline G_p$ and the progression $G_p$ itself. The centres of the clouds of $\mathbb Q_+$ and $\mathbb R_+$, endowed with the maximal metric $\varrho_1$ contain $\mathfrak{c}=2^{\aleph_0}$ and $2^\mathfrak{c}=2^{2^{\aleph_0}}$ complete spaces, respectively (see [5], Theorems 27 and 29). It was recently shown in [5] that the stabilizer of a strongly sparse set is minimal possible: it is trivial. We show below that the stabilizer of the cloud of a geometric progression is the minimal possible (nontrivial) subgroup of the group of similarities $(\mathbb R_+,\times )$. It is known that each nontrivial closed proper subgroup of $(\mathbb R_+,\times )$ is $G_p$ for some $p>1$. Theorem 2.5. For a metric $\varrho \in \mathcal{M}_p$ the following conditions are equivalent: 1) $\varrho \in \mathcal{IM}_p$; 2) the metric spaces $(X_p,p\varrho )$ and $(X_p,\varrho )$ are isometric; 3) $p\in \operatorname{St}_0(X_p,\varrho)$. Taking Theorem 2.4 into account we can state condition 2) as the equality $\operatorname{St}(X_p,\varrho )=G_p$, and we can state 3) as the equality $\operatorname{St}_0(X_p,\varrho)=G_p$. Theorem 2.6. For $\varrho \in \mathcal{M}_p$ the following conditions are equivalent: 1) $\operatorname{St}[(X_p,\varrho )]=G_p$; 2) $d_{\mathrm{GH}}\bigl((X_p,\varrho ),(X_p,p\varrho )\bigr)<\infty $; 3) there exists a metric $\rho \in \mathcal{IM}_p$ such that $d_{\mathrm{GH}}\bigl((X_p,\varrho),(X_p,\rho )\bigr)<\infty $. Using the next theorem we can construct examples of normalized metrics on a geometric progression which have smaller stabilisers. Theorem 2.7. For a positive integer $k$ and a metric $\varrho \in \mathcal{M}_p$ the following conditions are equivalent: 1) $p^k\in \operatorname{St}[(X_p,\varrho )]$; 2) there exist numbers $n_0$ and $M>0$ such that
$$
\begin{equation*}
\bigl|\varrho (p^{m+k},p^{n+k})-p^{k}\varrho (p^{m},p^{n})\bigr|<M
\end{equation*}
\notag
$$
for all integers $m,n\geqslant n_0$; 3) there exists $M>0$ such that for all integers $m$ and $n$.It is convenient to describe invariant metrics $\varrho \in \mathcal{M}_p$ in terms of a function $\alpha $. Proposition 2.8. The following conditions on a metric $\varrho \in \mathcal{M}_p$ are equivalent: 1) $\varrho \in \mathcal{IM}_p$; 2) there exists a function $\alpha \colon N\to \mathbb R$ such that If the set $X\subset [0,\infty )$ is clear from the context, then we let $\varrho $ be a metric in $\mathcal M_X$. For two metrics $\varrho $ and $\rho $ on a fixed set $X$ we let $d_{\mathrm{GH}}(\varrho,\rho )$ denote $d_{\mathrm{GH}}((X,\varrho ),(X,\rho ))$. It can be shown that metrics in $\mathcal{IM}_p$ have the property of ‘rigidity’: the existence of an embedding with finite distortion implies isometry. Theorem 2.9. Let $R$ be a complete relation between two metrics $\varrho,\rho \in \mathcal{IM}_p$ such that $\operatorname{dis} R<\infty $. Then $\varrho =\rho $. Corollary 2.10. The following conditions on two metrics $\varrho,\rho \in\mathcal{IM}_p$ are equivalent: 1) $\varrho =\rho $; 2) $d_{\mathrm{GH}}(\varrho,\rho )<\infty $. Theorem 2.11. For numbers $p,q>1$ and $\lambda,\mu >0$ and for metrics $\varrho \in \mathcal{IM}_p$ and $\rho \in \mathcal{IM}_q$ the following conditions are equivalent: 1) $p=q$, $\lambda =\mu p^k$ for some integer $k\in \mathbb Z$ and $\varrho =\rho $; 2) the metric spaces $(X_p,\lambda \varrho )$ and $(X_q,\mu \rho )$ are isometric; 3) $d_{\mathrm{GH}}((X_p,\lambda \varrho ),(X_q,\mu \rho ))<\infty $. In particular, this has two corollaries. Corollary 2.12. The following equality holds for any numbers $p>1$ and $\lambda >0$ and any metric $\varrho \in \mathcal{IM}_p$: Corollary 2.13. For numbers $p,q>1$ and $\lambda,\mu >0$ and metrics $\varrho \in \mathcal{IM}_p$ and $\rho \in \mathcal{IM}_q$ the following conditions are equivalent: 1) $p=q$ and $\varrho =\rho $; 2) the metric spaces $(X_p,\lambda \varrho )$ and $(X_q,\mu \rho )$ lie in the same orbit of the action of the group of similarities $(\mathbb R_+,\times )$. Here the orbit can be understood in the most rough sense of an action on clouds. The criterion of a metric obtained below (see Proposition 3.5) allows us to produce, on the basis of a geometric progression, examples of metrics with small (or even trivial) stabilizers; this shows, in particular, that some metrics $\varrho \in \mathcal{M}_p$ lie at an infinite distance from the set $\mathcal{IM}_p$. Recall (Corollary 2.10) that the Gromov-Hausdorff distance between different metrics in $\mathcal{IM}_p$ is infinite. Theorem 2.14. Let $-1\leqslant a\leqslant b\leqslant 1$ and $b\leqslant 1+2a$. Then for each function $\alpha \colon X\times X\to [a,b]$ the formula defines a normalized metric on $X$. Corollary 2.15. For each function $\alpha \colon X\times X\to\bigl[-\frac13,\frac13\bigr]$ the formula defines a normalized metric on $X$. Corollary 2.16. For each function $\alpha \colon X\times X\to [0,1]$ the formula defines a normalized metric on $X$. Theorem 2.17. Let $-1\leqslant a\leqslant b\leqslant 1$ and $b\leqslant \min\bigl\{1+\frac{2}{p}a,p+(p+1)a\bigr\}$. Then for each function $\alpha \colon X_p\times X_p\to [a,b]$ the formula defines a normalized metric on $X_p$. Corollary 2.18. For each function $\alpha \colon X_p\times X_p\to \bigl[-\frac{p}{p+2},\frac{p}{p+2}\bigr]$ the formula defines a normalized metric on $X_p$. Corollary 2.19. For each function $\alpha \colon \mathbb N\to \bigl[-\frac{p}{p+2},\frac{p}{p+2}\bigr]$ the formula defines a normalized metric on $X_p$. Corollary 2.20. For each function $\alpha \colon \mathbb N\to [0,1]$ the formula defines a normalized metric on $X_p$. Corollaries 2.16 and 2.20 describe infinite-dimensional convex sets of metrics from $\mathcal{IM}_p$. Corollary 2.21. For $p>1$ and $k\in \mathbb N$ there exists a metric $\varrho^{(k)}\in \mathcal{M}_p$ such that Corollary 2.21 shows that it also makes sense to consider normalized metrics with relaxed invariance property on a geometric progression. We say that a metric $\varrho \in \mathcal{M}_p$ is $k$-invariant normalized if
$$
\begin{equation*}
\varrho (p^m,p^n)=p^k\varrho (p^{m-k},p^{n-k}) \quad\text{for all } n,m\in \{-\infty \}\cup \mathbb Z.
\end{equation*}
\notag
$$
We denote the set of $k$-invariant normalized metrics on $X_p$ by $\mathcal{IM}_p^{(k)}$. It is obvious that $\mathcal{IM}_p\subset \mathcal{IM}_p^{(k)}\subset \mathcal{IM}_p^{(kr)}$ for all $k,r\in \mathbb N$. Many results on metrics in $\mathcal{IM}_p$ can be transferred to $\mathcal{IM}_p^{(k)}$. Theorem 2.22. For a metric $\varrho \in \mathcal{M}_p$ the following conditions are equivalent: 1) $\varrho \in \mathcal{IM}_p^{(k)}$; 2) the metric spaces $(X_p,p^k\varrho )$ and $(X_p,\varrho )$ are isometric; 3) $p^k\in \operatorname{St}_0(X_p,\varrho)$. Taking Theorem 2.4 into account we can state condition $2)$ as the inclusion $G_{p^k}\subset \operatorname{St}(X_p,\varrho)$ and condition $3)$ as the inclusion $G_{p^k}\subset \operatorname{St}_0(X_p,\varrho)$. Corollary 2.23. For metrics $\varrho,\rho \in \mathcal{IM}_p^{(k)}$ the following conditions are equivalent: 1) $\varrho =\rho $; 2) $d_{\mathrm{GH}}(\varrho,\rho )<\infty $. It follows from the inclusions $\mathcal{IM}_p^{(k)},\mathcal{IM}_p^{(r)}\subset \mathcal{IM}_p^{(kr)}$ that the set of metrics $\mathcal{IM}_p^{(\infty )}=\bigcup_{i=1}^\infty \mathcal{IM}_p^{(i)}\subset \mathcal{M}_p$ is convex, and two arbitrary metrics in it lie at an infinite distance. This is precisely the set of normalized metrics in $X_p$ such that the stabilizer $\operatorname{St}_0$ is nontrivial. The representation $\mathbb N=\bigsqcup_{r=0}^{k-1}(r+k\mathbb N)$ yields the natural maps
$$
\begin{equation*}
\Pi \colon \mathcal{M}_p\to (\mathcal{M}_p)^{k}\quad\text{and} \quad \Pi_k\colon \mathcal{IM}_p^{(k)}\to (\mathcal{IM}_p)^{k},
\end{equation*}
\notag
$$
which are not epimorphisms. The problem of combining $k$ particular metrics into a single one can be solved using Propositions 3.5 and 3.6. We stress that such a metric has an infinite distance to the set $\mathcal{IM}_p^\infty $. One can describe normalized (Proposition 3.5) and invariant normalized (Proposition 3.6) metrics in terms of a function $\alpha $, which enables one to give a constructive description of a wider class of invariant metrics than (4), when the constant $\alpha $ is replaced by an exponential function. The next theorem provides a number of new examples of invariant normalized metrics on a geometric progression. Theorem 2.25. The function is a metric in $X_p$ if and only if For $\delta =0$ we see that the condition $-1\leqslant \alpha \leqslant 1$ is not only necessary but also sufficient in (4). For different $\delta $ the functions $p^{-\delta i}$ are linearly independent, so Theorem 2.25 also shows that the convex set $\mathcal{IM}_p$ is infinite-dimensional. Corollary 2.26. For numbers 1) $p=q$, $\delta =\theta $, $\alpha =\beta $ and $\lambda =\mu p^k$ for some integer $k\in \mathbb Z$; 2) the metric spaces $(X_p,\lambda \varrho_{\delta,\alpha})$ and $(X_q,\mu \varrho_{\theta,\beta})$ are isometric; 3) $d_{\mathrm{GH}}\bigl((X_p,\lambda \varrho_{\delta,\alpha}),(X_q,\mu \varrho_{\theta,\beta})\bigr)<\infty $. Corollary 2.27. For numbers 1) $p=q$, $\delta =\theta $ and $\alpha =\beta $; 2) the metric spaces $(X_p,\lambda \varrho_{\delta,\alpha})$ and $(X_q,\mu \varrho_{\theta,\beta})$ lie in the same orbit of the action of the group of similarities $(\mathbb R_+,\times )$. § 3. ProofsThe technical results presented below are more or less known. However, the most complicated of them have only been published for subsets of the real line parametrized by positive integers and, formally, cannot be applied to subsets parametrized by integers, so we state and prove them for completeness. These are in fact results on estimates for the ‘displacement’ of an arbitrary point (or the reference point $0_X$ alone) under a correspondence with prescribed finite distortion. Proposition 3.1. Given a complete relation $R$ between spaces $X_{\varrho_X}$ and $Y_{\varrho_Y}$, if $\operatorname{dis} R<M$, then the inequality $|x-y_x|<K$ holds for any points $x\in X$ and $y_x\in R(x)$, where $K=M+y_0$. Proof. The condition $\operatorname{dis} R<M$ means that
$$
\begin{equation*}
\sup\bigl\{|\varrho_X(x,x')-\varrho_Y(y_x,y_{x'})|\bigr\}<M.
\end{equation*}
\notag
$$
We can write the above as the two inequalities
$$
\begin{equation}
\varrho_Y(y_x,y_{x'})-M<\varrho_X(x,x')<\varrho_Y(y_x,y_{x'})+M
\end{equation}
\tag{5}
$$
for arbitrary points $x,x'\in X$, $y_x\in R(x)$ and $y_{x'}\in R(x')$. We write (5) for the fixed point $x'=0$, taking (2) into account: These inequalities mean that $|x-y_x|<y_{0}+M$.
The proof is complete. Corollary 3.2. Given a complete relation $R$ between spaces $X_{\varrho_X}$ and $Y_{\varrho_Y}$, if ${\operatorname{dis} R<M}$, then $X\subset U_K(Y)$, where $K=M+y_0$. Here $U_K(Y)$ denotes the open neighbourhood of radius $K$ in the standard metric $\varrho_{-1}$ on the real line. Proof of Theorem 2.1. Let $R\subset X\times Y$ be a correspondence between metric spaces $(X_p,\varrho_X)$ and $(X,\varrho_Y)$ such that $\operatorname{dis} R<\infty $. Fix $M>0$ such that $\operatorname{dis} R<M$. By Corollary 3.2, for $K_1=M+y_0$ and $K_2=M+x_0$ such that $x_0\in R^{-1}(0)=R^*(0)$ we have the inclusions
The proof is complete. Proposition 3.3. Given a complete relations $R$ between metrics $\varrho,\rho \in \mathcal{M}_p$, if $\operatorname{dis} R<M$, then Proof. For a point $y\in G_p$, if $y\ne y_0$, then
Consider a point $x\in X_p$ such that $x\leqslant \frac{p-1}{p}y_0-M$. Then and therefore $y_x=y_0$.Next consider $x\in X_p$ such that $x\geqslant M$. Then so that $y_x\ne y_0$.Thus we have shown that the set $\bigl(0,\frac{p-1}{p}y_0-M\bigr]\cap [M,\infty )$ cannot contain points from $X_p$. This does not mean that $\frac{p-1}{p}y_0-M<M$: this restriction on $y_0$ is in fact equivalent to the absence of an integer $n$ such that
$$
\begin{equation*}
\log_p \biggl(\frac{p-1}{p}y_0-M\biggr)\geqslant n\geqslant \log_p M.
\end{equation*}
\notag
$$
This latter condition implies that Exponentiating we obtain $(p-1)y_0-pM<M$, that is, $y_0<\frac{p+1}{p-1}M$.
The proof if complete. Proposition 3.4. Given a complete relation $R$ between metrics $\varrho,\rho \in \mathcal{M}_p$, if $\operatorname{dis} R<M$, then $R(x)=x$ for all $x\in X_p$ such that Proof. By Propositions 3.1 and 3.3, for all $x\in X_p$ and $y_x\in R(x)$ we have For $x'\ne x$ we have the estimate If $x\geqslant \frac{2p^2}{(p-1)^2}M$ and $y_x\ne x$, then we obtain
$$
\begin{equation*}
|x-y_x|\geqslant \frac{p-1}{p}x\geqslant \frac{2p}{p-1}M.
\end{equation*}
\notag
$$
This inequality is reverse to (6). Hence $y_x= x$.
The proof is complete. Proof of Theorem 2.2. For all $n\geqslant 1$ fix correspondences $R_n\in \mathcal R(X_p,Z)$ such that $\operatorname{dis} R_n<\frac{(p-1)^2}{2p^2n}$. Consider the correspondence $R=R_m^*\circ R_n\in \mathcal R(X_p,X_p)$. By the triangle inequality
$$
\begin{equation*}
\operatorname{dis} R\leqslant \operatorname{dis} R_n+\operatorname{dis} R_m<\frac{(p-1)^2}{2p^2}\frac{n+m}{nm}.
\end{equation*}
\notag
$$
By Proposition 3.4, for $x\in X_p$ it follows from $x\geqslant \frac{n+m}{nm}$ that $R(x)=x$. This means that
Setting $x=x'$ in the definition (1) of distortion we see that for each correspondence $R\in \mathcal R(X,Z)$ and any point $x\in X$ the set $\operatorname{diam}R(x)$ has diameter at most $\operatorname{dis} R$. Therefore, $\operatorname{diam}R_m(x)\leqslant \frac{(p-1)^2}{2p^2m}$. Letting $m$ tend to infinity in (7), for $x\in X_p$ we see that, if $x\geqslant \frac1{n}$, then $R_n(x)$ is a one-point set and $R_n(x)=R_m(x)=y_x$ for all ${m\geqslant n}$. Thus, a complete correspondence between $X_p\setminus \{0\}$ and $Z$ is well defined. For points $x,x'\in X_p\setminus \{0\}$ consider a positive integer $n$ such that $x,x'\geqslant \frac{1}{n}$. Then
$$
\begin{equation*}
\bigl|\varrho (x,x')-\varrho_Z(y_x,y_{x'})\bigr|<\operatorname{dis} R_n<\frac{(p-1)^2}{2p^2n}.
\end{equation*}
\notag
$$
Letting $n$ tend to infinity we obtain $\varrho (x,x')=\varrho_Z(y_x,y_{x'})$ for all $x,x'>0$. Defining the relation (map) $R$ at $0$ by $R(0)=\lim_{n\to -\infty}R(p^n)$ we obtain the required isometry.
The proof is complete. Proof of Theorem 2.4. Let $\lambda \in \operatorname{St}[(X_p,\varrho )]$. By Proposition 1.4
$$
\begin{equation*}
d_{\mathrm{GH}}\bigl((\lambda X_p,\varrho_{\lambda X_p}),(X_p,\varrho_{X_p})\bigr)=d_{\mathrm{GH}}\bigl((X_p,\lambda \varrho_{X_p}),(X_p,\varrho_{X_p})\bigr)<\infty .
\end{equation*}
\notag
$$
By Theorem 2.1 Hence $\lambda =\mu p^k$ for some $k\in \mathbb Z$, that is, $\lambda \in G_p$.
The proof is complete. Proof of Theorem 2.5. $1)\Rightarrow 2)$. By Proposition 1.4 the spaces $(X_p,p\varrho_{X_p})$ and $(pX_p,\varrho_{pX_p})$ are isometric. On the other hand $pX_p=X_p$ and $\varrho_{pX_p}=\varrho $ by the assumption that $\varrho \in \mathcal{IM}_p$.
The implication $2)\Rightarrow 3)$ is obvious: $p\in \operatorname{St}(X_p,\varrho)\subset \operatorname{St}_0(X_p,\varrho)$. $3)\Rightarrow 1)$. Consider arbitrary points $x=p^m$ and $x'=p^n\in X_p$, and let $\varepsilon >0$ be such that Let $R$ be a correspondence such that $\operatorname{dis} R_{p\varrho,\varrho}<\varepsilon $. By Proposition 1.4, in place of the metric $p\varrho $ on $X_p$ we can consider the metric $\varrho_{pX_p}$ on $pX_p = X_p$. Since ${\varrho_{pX_p}\in \mathcal{M}_p}$, we can use Proposition 3.4. Hence $R(x) = x$ and $R(x')=x'$. Therefore, It follows that $\varrho (x,x')-\varrho_{pX_p}(x,x')=0$ for any points $x,x'\in X_p$. Recall the definition of $\varrho_{pX_p}$. Then for all $x,x'\in X_p$ we obtain
$$
\begin{equation*}
\varrho (x,x')=\varrho_{pX_p}(x,x')=p\varrho (p^{-1}x,p^{-1}x'),
\end{equation*}
\notag
$$
that is, $\varrho \in \mathcal{IM}_p$.
The proof is complete. Proof of Theorem 2.6. Taking Theorem 2.4 into account, the equivalence $1)\Leftrightarrow 2)$ is just the definition of the cloud stabilizer $\operatorname{St}[(X_p,\varrho )]$.
The implication $3)\Rightarrow 2)$ follows from the chain of inequalities
$$
\begin{equation*}
\begin{aligned} \, d_{\mathrm{GH}}(p\varrho,\varrho ) &\leqslant d_{\mathrm{GH}}(p\varrho,p\rho )+d_{\mathrm{GH}}(p\rho,\rho )+d_{\mathrm{GH}}(\rho,\varrho ) \\ &=d_{\mathrm{GH}}(p\varrho,p\rho )+d_{\mathrm{GH}}(\rho,\varrho )= (p+1)d_{\mathrm{GH}}(\varrho,\rho )<\infty . \end{aligned}
\end{equation*}
\notag
$$
$2)\Rightarrow 3)$. Consider a new metric on $X_p$:
$$
\begin{equation}
\rho (p^m,p^n)=\lim_{k\to \infty}p^{-k}\varrho (p^{m+k},p^{n+k}).
\end{equation}
\tag{8}
$$
We claim that a finite limit exists in (8). For a proof we show that the numbers $x_k=x_k(m,n)=p^{-k}\varrho (p^{m+k},p^{n+k})$ form a Cauchy sequence. Let $R$ be a correspondence such that $\operatorname{dis} R_{\varrho,p\varrho}<M<\infty $. In view of Proposition 1.4, in place of the metric $p^{-k}\varrho (p^{m},p^{n})$ на $X_p$ we can consider $\varrho_{p^{-k}X_p}=p^{-k}\varrho (p^{m+k},p^{n+k})\in \mathcal{M}_p$ on $p^{-k}X_p=X_p$. By Proposition 3.4, for any $n<m$ there exists $k_0$ such that $R(p^{m+k})=p^{m+k}$ and $R(p^{n+k})=p^{n+k}$ for all $k\geqslant k_0$. Therefore,
$$
\begin{equation*}
\bigl|\varrho (p^{m+k},p^{n+k})-p^{-1}\varrho (p^{m+k+1},p^{n+k+1})\bigr|<M
\end{equation*}
\notag
$$
for all $k\geqslant k_0$. Now, for $k\geqslant k_0$ and $r\geqslant 1$ we find an estimate for the difference $|x_k-x_{k+r}|$:
$$
\begin{equation}
\begin{aligned} \, \notag |x_{k}-x_{k+r}| &=\biggl|\sum_{i=0}^{r-1}(x_{k+i}-x_{k+i+1})\biggr|\leqslant \sum_{i=0}^{r-1}|x_{k+i}-x_{k+i+1}| \\ \notag &=p^{-k}\sum_{i=0}^{r-1}p^{-i}\bigl|\varrho (p^{m+k+i},p^{n+k+i})-p^{-1}\varrho (p^{m+k+i+1},p^{n+k+i+1})\bigr| \\ &<p^{-k}\sum_{i=0}^{r-1}p^{-i}M=p^{-k}\frac{1-p^{-r}}{1-p^{-1}}M<p^{-k}\frac{1}{1-p^{-1}}M=p^{-k}\frac{p}{p-1}M. \end{aligned}
\end{equation}
\tag{9}
$$
Hence for sufficiently large $k$ the difference $|x_{k}-x_{k+r}|$ can be made arbitrarily small for all $r\geqslant 1$.
It is obvious that the limit values $\rho (p^m,p^n)$ define a metric such that $\rho \in \mathcal{M}_p$. It is also obvious that $\rho (p^{m+1},p^{n+1})=p^{-1}\rho (p^m,p^n)$, that is, $\rho \in \mathcal{IM}_p$. It follows from (9) that
$$
\begin{equation*}
\bigl|\varrho (x,x')-\rho (x,x')\bigr|<\frac{p}{p-1}M<\infty
\end{equation*}
\notag
$$
for all $x>x'\geqslant \frac{2p^2}{(p-1)^2}M$. For $x>x'<\frac{2p^2}{(p-1)^2}M$ we have the inequalities
$$
\begin{equation*}
\begin{gathered} \, x-\frac{2p^2}{(p-1)^2}M<x-x'\leqslant \varrho (x,x')\leqslant x+x'<x+\frac{2p^2}{(p-1)^2}M, \\ x-\frac{2p^2}{(p-1)^2}M<x-x'\leqslant \rho (x,x')\leqslant x+x'<x+\frac{2p^2}{(p-1)^2}M, \end{gathered}
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\bigl|\varrho (x,x')-\rho (x,x')\bigr|<2\frac{2p^2}{(p-1)^2}M<\infty .
\end{equation*}
\notag
$$
The proof is complete. Proof of Theorem 2.7. The implication $3)\Rightarrow 1)$ is obvious: it is sufficient to set $R(x)=p^{-k}x$ for all $x\in X_p$.
$1)\Rightarrow 2)$. Let $d_{\mathrm{GH}}\bigl((X_p,p^k\varrho_{X_p}),(X_p,\varrho_{X_p})\bigr)<\infty$. By Proposition 1.4, $d_{\mathrm{GH}}\bigl((X_p, \varrho_{X_p}), (p^kX_p,\varrho_{p^kX_p})\bigr)<\infty$. Let $R$ be a correspondence between the latter two metric spaces such that $\operatorname{dis} R<M$ for some $M<\infty $. By Proposition 3.4 there exists an integer $n_0$ such that $R(p^n)=p^n$ for all $n\geqslant n_0$. Then
$$
\begin{equation*}
\bigl|\varrho (p^{m+k},p^{n+k})-p^{k}\varrho (p^{m},p^{n})\bigr|<M \quad\text{for all } m>n\geqslant n_0.
\end{equation*}
\notag
$$
$2)\Rightarrow 3)$. If $n<n_0$, then it follows from the inequalities
$$
\begin{equation*}
p^{m+k}-p^{n_0+k}<p^{m+k}-p^{n+k}\leqslant \varrho (p^{m+k},p^{n+k})\leqslant p^{m+k}+p^{n+k}<p^{m+k}+p^{n_0+k}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
p^{m+k}-p^{n_0+k}<p^{m+k}-p^{n+k}\leqslant \varrho (p^{m+k},p^{n+k})\leqslant p^k(p^{m}+p^{n})<p^{m+k}+p^{n_0+k}
\end{equation*}
\notag
$$
that
$$
\begin{equation*}
\bigl|\varrho (p^{m+k},p^{n+k})-p^{k}\varrho (p^{m},p^{n})\bigr|<2p^{n_0} \quad\text{for all } m>n<n_0.
\end{equation*}
\notag
$$
Clearly, each pair $m$, $n$ is either as above or among the ones considered in condition $2)$.
The proof is complete. Proof of Proposition 2.8. The implication $2)\Rightarrow 1)$ is obvious.
$1)\Rightarrow 2)$. Consider the function $\alpha (r)=\varrho (p^r,1)-p^r$, $r\geqslant 1$. Given a metric $\varrho \in \mathcal{IM}_p$, using induction on $n$ we can prove that
$$
\begin{equation*}
\varrho (p^m,p^n)=p^n\varrho (p^{m-n},p^0)=p^n(p^{m-n}+\alpha (m-n))=p^m+\alpha (m-n)p^n.
\end{equation*}
\notag
$$
The proof is complete. Proof of Theorem 2.9. By Proposition 3.4, for some $N_0$ and each $m\geqslant N_0$ we have $R(p^m)=p^m$. Hence for all $m,n\geqslant N_0$ we have the inequality $\bigl|\varrho (p^m,p^n)-\rho (p^m,p^n)\bigr|<M$. By Proposition 2.8 for some functions $\alpha_1,\alpha_2\colon \mathbb N\to \mathbb R$. Thus, for all $m,n\geqslant N_0$ we have the inequality $\bigl|\alpha_1(m-n)-\alpha_2(m-n)\bigr|p^n<M$. Fixing $j=m- n$ and letting $n$ tend to infinity, we obtain $\alpha_1(j)-\alpha_2(j)=0$ for each $j\geqslant 1$.
The proof is complete. Proof of Corollary 2.10. The implication $1)\Rightarrow 2)$ is obvious.
$2)\Rightarrow 1)$. Let $R$ be a correspondence such that $\operatorname{dis} R<\infty $. By Theorem 2.9 ${\varrho =\rho}$. The proof is complete. Proof of Theorem 2.11. The implications $1)\Rightarrow 2)\Rightarrow 3)$ are obvious.
$3)\Rightarrow 1)$. By Proposition 1.4
$$
\begin{equation*}
d_{\mathrm{GH}}\bigl((\lambda X_p,\varrho_{\lambda X_p}),(\mu X_q,\rho_{\mu X_q})\bigr)=d_{\mathrm{GH}}\bigl((X_p,\lambda \varrho ),(X_q,\mu \rho )\bigr)<\infty.
\end{equation*}
\notag
$$
By Theorem 2.1 $d_{\mathrm{GH}}\bigl((\lambda X_p,\varrho_{-1}),(\mu X_q,\varrho_{-1})\bigr)<\infty $. Hence $p=q$ and $\lambda =\mu p^k$ for some $k\in \mathbb Z$. By Corollary 2.10 $\varrho =\rho $.
The proof is complete. Proof of Corollary 2.13. The implication $1)\Rightarrow 2)$ is obvious.
$2)\Rightarrow 1)$. Since the metric spaces $(X_p,\lambda \varrho )$ and $(X_q,\mu \rho )$ lie in the same orbit of the action of the similarity group $\mathbb R_+,\times )$, there exists a number $\nu $ such that the spaces $(X_p,\nu \lambda \varrho )$ and $(X_q,\mu \rho )$ lie in the same cloud. By Corollary 2.11 $p=q$ and ${\varrho =\rho}$. The proof is complete. Proposition 3.5. Given a function $\alpha \colon X\times X\to \mathbb R$, the formula defines a normalized metric if and only if for all $x''<x'<x$: Proof. We verify inequality 0). We write down conditions ensuring that the function $\varrho_\alpha (x,x')=x+\alpha (x,x')x'$ satisfies the triangle inequality. Relations (3) (the triangle inequality for $x''=0$) take the form Hence $-1\leqslant \alpha(x,x')\leqslant 1$. These inequalities imply that the function $\varrho_\alpha (x,x')$ is positive. Since $\varrho_\alpha (x,0)$ is independent of the values of $\alpha (x,0)$, we cannot obtain constraints on these values. However, the normalization condition holds, so we can assume that the required inequalities $-1\leqslant \alpha (x,0)=\alpha (0,x)\leqslant 1$ are true.
We prove I). The triangle inequality for the pair $(x,x'')$ looks as follows:
$$
\begin{equation*}
x+\alpha(x,x'')x''\leqslant x+\alpha(x,x')x'+x'+\alpha(x',x'')x'',
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\bigl(\alpha(x,x'')-\alpha(x',x'')\bigr)x''\leqslant x'+\alpha(x,x')x'.
\end{equation*}
\notag
$$
We prove II). We write the triangle inequality for the pair $(x,x')$:
$$
\begin{equation*}
x+\alpha(x,x')x'\leqslant x+\alpha(x,x'')x''+x'+\alpha(x',x'')x'',
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\alpha(x,x')x'-x'\leqslant \bigl(\alpha(x,x'')+\alpha(x',x'')\bigr)x''.
\end{equation*}
\notag
$$
We prove III). We write the triangle inequality for the pair $(x',x'')$:
$$
\begin{equation*}
x'+\alpha(x',x'')x''\leqslant x+\alpha(x,x')x'+x+\alpha(x,x'')x'',
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\bigl(\alpha(x',x'')-\alpha(x,x'')\bigr)x''\leqslant 2x-x'+\alpha(x,x')x'.
\end{equation*}
\notag
$$
Thus, we have expressed all the properties of the metric $\varrho_\alpha $ in terms of the function $\alpha (x,x')$. The proof is complete. Proof of Theorem 2.14. We write inequalities I)–III) for the possible values of $\alpha $ and positions of the points $0\leqslant x''<x'<x$ for which they look most precarious. In I) we replace the left-hand side by its maximum value $(b-a)x''$ and the right-hand side by its minimum value $(1+a)x'$. Bearing in mind that $x''<x'$ we obtain the valid inequality $(b-a)x''\leqslant (1+a)x'$.
In II) we replace the first term on the right-hand side by its minimum value $2ax''$; the minimum value of the second term is $(1 - b)x'$. If $a\geqslant 0$, then II) is obvious. Bearing in mind that $x''<x'$ we obtain the valid inequality $0<2ax''+(1-b)x'$ for negative $a$ too. In III) we replace the left-hand side by its maximum value $(b-a)x''$ and the right-hand side by its minimum value $2x-x'+ax'$. Bearing in mind that $x''<x'<x$ we obtain the valid inequality $(b-a)x''\leqslant x'+ax'$. The proof is complete. Proof of Theorem 2.17. We write inequalities I)–III) for the possible values of $\alpha $ and $1<n<m$ for which they look most precarious. In I) we replace the left-hand side by its maximum value $(b-a)p^l$ and the right-hand side by its minimum value $(1+a)p^n$. Bearing in mind that $l+1\leqslant n$ we obtain the valid inequality $(b-a)p^l\leqslant (1+a)p^{l+1}$.
In II) we replace the first term on the right-hand side by the minimum value $2ap^l$; the minimum value of the second is $(1- b)p^n$. If $a\geqslant 0$, then inequality II) is obvious. Bearing in mind that $l+1\leqslant n$ we obtain the valid inequality $0\leqslant 2ap^l+(1-b)p^{l+1}$ for negative $a$ too. In III) we replace the left-hand side by its maximum value $(b-a)p^l$ and the right-hand side by its minimum value $2p^m - (1 - a)p^n$. Bearing in mind that $l<n< m$, for $m=n+1$ we obtain the strongest inequality making the strongest assumption $n=l+1$, we reduce it to $b-a\leqslant 2p^2-p+pa$. The last inequality holds becauseThe proof is complete. Proposition 3.6. For a function $\alpha \colon \mathbb N\to \mathbb R$ the formula defines an (invariant) normalized metric if and only if the following inequalities hold for all $i,j\in \mathbb N$: Proof. Recall that we always assume that $l<n<m$. We also introduce the notation $i=n-l\geqslant 1$ and $j=m-n\geqslant 1$.
We write conditions 0)–III) in Proposition 3.5 for the points $x'' = p^l$, $x' = p^n$ and $x = p^m$. Reducing by $p^l$ we obtain the required conditions $\mathrm{0_I})$–$\mathrm{III_I})$. The proof is complete. Proof of Corollary 2.21. The case $k=1$ can be realized by any metric in $\mathcal{IM}_p$, so assume that $k\geqslant 2$. Consider the function We claim that $\varrho_\alpha (p^m,p^n)=p^m+\alpha(m,n)p^n$ is the required metric. It is easy to see that $\varrho_\alpha (p^{m+k},p^{n+k})=p^k\varrho_\alpha (p^m,p^n)$, so $p^k\in \operatorname{St}(X,\varrho_\alpha )\subset \operatorname{St}[(X,\varrho_\alpha )]$.
By Theorem 2.4, $\operatorname{St}[(X_p,\varrho_\alpha )]\subset G_p$. Hence it is sufficient to show that no number $p^r$ for $1\leqslant r<k$ lies in the stabilizer. We estimate $\bigl|\varrho (p^{m+r},p^{n+r})-p^{r}\varrho (p^{m},p^{n})\bigr|$ for $1\leqslant r<k$. The integers $m$ and $m+r$ cannot both be multiples of $k$. Hence for $m\equiv 0\mod k$ we have the equality
$$
\begin{equation*}
\bigl|\varrho (p^{m+r},p^{n+r})-p^{r}\varrho (p^{m},p^{n})\bigr|=\bigl|p^{m+r}-p^{r}(p^{m}+p^{n})\bigr|=p^{n+r} \xrightarrow{n\to \infty}\infty .
\end{equation*}
\notag
$$
By Theorem 2.7, $p^r\notin \operatorname{St}[(X_p,\varrho_\alpha )]$ for $1\leqslant r<k$.
The proof is complete. Proof of Corollary 2.24. Consider the function We claim that $\varrho_\alpha (p^m,p^n)=p^m+\alpha(m,n)p^n$ is the required metric.
We estimate $\bigl|\varrho (p^{m+k},p^{n+k})-p^{k}\varrho (p^{m},p^{n})\bigr|$ for $k\geqslant 1$:
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\varrho (p^{m+k},p^{n+k})-p^{k}\varrho (p^{m},p^{n})\bigr| \\ &\qquad=\biggl|p^{m+k}+\biggl(1-\frac{1}{n+k}\biggr)p^{n+k}-p^{k} \biggl(p^{m}+\biggl(1-\frac{1}{n}\biggr)p^{n}\biggr)\biggr| \\ &\qquad = \biggl(\frac{1}{n}-\frac{1}{n+k}\biggr)p^{n+k}=\frac{k}{n(n+k)}p^{n+k}\xrightarrow{n\to \infty}\infty. \end{aligned}
\end{equation*}
\notag
$$
By Theorem 2.7, $p^k\notin \operatorname{St}[(X_p,\varrho_\alpha )]$.
The proof is complete. Proof of Theorem 2.25. We write conditions $\mathrm{0_I})$–$\mathrm{III_I})$ from Proposition 3.6 for the function $\alpha (j)=\alpha p^{-\delta j}$, where $\alpha $ is a constant. Inequality $\mathrm{0_I})$ means that for each $j\geqslant 1$ we have $-1\leqslant \alpha p^{-\delta j}\leqslant 1$. Assume that $\delta <0$. Letting $j$ tend to $\infty $ we obtain that $p^{-\delta j}$ also tends to $\infty $. Hence $\alpha p^{-\delta j}$ tends to $\operatorname{sign}(\alpha )\cdot \infty $, and thus leaves the unit interval $[-1,1]$. Therefore, $\delta \geqslant 0$. The integer $j$ varies from $1$ to $\infty $. Hence $\mathrm{0_I})$ is equivalent to the inequality that is, $-p^{\delta}\leqslant \alpha \leqslant p^{\delta}$.
Inequality $\mathrm{I_I})$ assumes the form
$$
\begin{equation*}
\alpha p^{-\lambda r-\lambda \delta}-\alpha p^{-\lambda r}\leqslant p^{r}+\alpha p^{r-\lambda \delta},
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\alpha \bigl[p^{-\lambda r-\lambda \delta}-p^{-\lambda r}-p^{r-\lambda \delta}\bigr]\leqslant p^{r}.
\end{equation*}
\notag
$$
We find the maximum of the positive numbers $\Delta(r,\delta )=p^{r-\lambda \delta}+p^{-\lambda r}-p^{-\lambda r-\lambda \delta}$. With growth of $\delta $ the quantity $\Delta=p^{-\lambda \delta}(p^{r}\,{-}\,p^{-\lambda r})\,{+}\,p^{-\lambda r}$ decreases (for fixed $r$), so that $\Delta(r,\delta )\leqslant \Delta(r,1)=p^{-\lambda}(p^{r}-p^{-\lambda r})+p^{-\lambda r}$. Now we expand the required inequality $-a\Delta(r,1)\leqslant p^r$:
$$
\begin{equation*}
a(p^{r-\lambda}-p^{-\lambda r-\lambda}+p^{-\lambda r})\geqslant -p^r,
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
a(p^{-\lambda}+p^{-(\lambda+1) r}(1-p^{-\lambda}))\geqslant -1.
\end{equation*}
\notag
$$
The positive number $p^{-\lambda}+p^{-(\lambda+1) r}(1-p^{-\lambda})$ decreases with growth of $r$, so we obtain the strongest condition on $a$ for $r=1$. Multiplying this strongest inequality
$$
\begin{equation*}
a(p^{-\lambda}+p^{-\lambda-1}-p^{-2\lambda -1})\geqslant -1
\end{equation*}
\notag
$$
by $p^{2\lambda +1}$, for $\alpha $ we obtain the lower bound from (5). Note that this inequality is stronger than the left-hand inequality in (10); it is precisely inequality I) for $r=\delta =1$.
Inequality II) assumes the following form:
$$
\begin{equation*}
\alpha p^{r-\lambda \delta}-p^{r}\leqslant \alpha p^{-\lambda r-\lambda \delta}+\alpha p^{-\lambda r},
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\alpha [p^{r-\lambda \delta}-p^{-\lambda r-\lambda \delta}-p^{-\lambda r}]\leqslant p^{r}.
\end{equation*}
\notag
$$
We find upper and lower bounds for the quantities $\Delta(r,\delta )=p^{r-\lambda \delta}-p^{-\lambda r-\lambda \delta}-p^{-\lambda r}$. Since $p^{r}-p^{-\lambda r}>0$, it follows that
$$
\begin{equation*}
\begin{aligned} \, -p^{-\lambda r} &=\Delta(r,\infty )<\Delta(r,\delta )=p^{-\lambda \delta}(p^{r}-p^{-\lambda r})-p^{-\lambda r}\leqslant \Delta(r,1) \\ &=p^{r-\lambda}-p^{-\lambda r-\lambda}-p^{-\lambda r}= p^{r-\lambda}-p^{-\lambda r}(1+p^{-\lambda})<p^{r-\lambda} \end{aligned}
\end{equation*}
\notag
$$
and the values of $\Delta(r,\delta )$ can be arbitrarily close to the lower and upper bounds (if $r$ is sufficiently large). Taking all these estimates into account, the main inequality $\alpha \Delta(r,\delta )\leqslant p^r$ assumes the following form:
$$
\begin{equation*}
-p^{(1+\lambda )r}\leqslant -p^{1+\lambda}\leqslant \alpha \leqslant p^{\lambda}.
\end{equation*}
\notag
$$
Combining all conditions on $\alpha $ obtained above results in the inequalities stated in the theorem. In the remaining case when the triangle inequality III) must be checked, it is rather difficult to produce effective conditions on the coefficient $\alpha $, so we show simply that, if we assume that the inequalities in the theorem that we have already established are valid, then the remaining configuration of three points satisfies the triangle inequality. Inequality III) takes the following form:
$$
\begin{equation*}
p^{r}-\alpha p^{r-\lambda \delta}\leqslant 2p^{r+\delta}+\alpha p^{-\lambda r-\lambda \delta}-\alpha p^{-\lambda r},
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\alpha [p^{-\lambda r}-p^{r-\lambda \delta}-p^{-\lambda r-\lambda \delta}]\leqslant 2p^{r+\delta}-p^{r}.
\end{equation*}
\notag
$$
We find estimates for $\Delta(r,\delta )=p^{-\lambda r}-p^{r-\lambda \delta}-p^{-\lambda r-\lambda \delta}$. Since $p^{r}+p^{-\lambda r}>0$, we have
$$
\begin{equation*}
\Delta(r,\delta )=p^{-\lambda r}-p^{-\lambda \delta}(p^{r}+p^{-\lambda r})<\Delta(r,\infty )=p^{-\lambda r}.
\end{equation*}
\notag
$$
The positive values of $\Delta(r,\delta )$ produce the upper bound $\alpha p^{-\lambda r}\leqslant 2p^{r+\delta}-p^{r}=p^{r}(2p^{\delta}-1)$, which holds for all $\alpha \leqslant p^{\lambda}$:
$$
\begin{equation*}
\alpha \Delta(r,\delta )\leqslant p^{\lambda}p^{-\lambda r}\leqslant 1<p^{r}(2p^{\delta}-1).
\end{equation*}
\notag
$$
Next we examine the lower bound for $\alpha $ for negative values of $\Delta(r,\delta )$. Thus, we look at the inequality $(-\alpha )\bigl(-\Delta(r,\delta )\bigr)\leqslant 2p^{r+\delta}-p^{r}$ for $r$ and $\delta $ such that $-\Delta(r,\delta )>0$. We show that
$$
\begin{equation}
\frac{p^{2\lambda +1}}{p^{\lambda +1}+p^{\lambda}-1}(p^{r-\lambda \delta}+p^{-\lambda r-\lambda \delta}-p^{-\lambda r})\leqslant 2p^{r+\delta}-p^{r}.
\end{equation}
\tag{11}
$$
As $\delta $ increases, the left-hand side decreases, while the right-hand side increases. Hence inequality (11) is in its strongest form for $\delta =1$:
$$
\begin{equation}
\frac{p^{2\lambda +1}}{p^{\lambda +1}+p^{\lambda}-1}(p^{r-\lambda}+p^{-\lambda r-\lambda}-p^{-\lambda r})\leqslant 2p^{r+1}-p^{r},
\end{equation}
\tag{12}
$$
that is,
$$
\begin{equation*}
\begin{gathered} \, p^{2\lambda +1}(p^{r-\lambda}+p^{-\lambda r-\lambda}-p^{-\lambda r})\leqslant (p^{\lambda +1}+p^{\lambda}-1)(2p^{r+1}-p^{r}), \\ \begin{aligned} \, &p^{r+\lambda +1}+p^{-\lambda r+\lambda +1}-p^{-\lambda r+2\lambda +1} \\ &\qquad\qquad\leqslant 2p^{r+\lambda +2}+2p^{r+\lambda +1}-2p^{r+1}-p^{r+\lambda +1}-p^{r+\lambda}+p^{r}, \end{aligned} \\ p^{-\lambda r+\lambda +1}-p^{-\lambda r+2\lambda +1}\leqslant 2p^{r+\lambda +2}-2p^{r+1}-p^{r+\lambda}+p^{r}. \end{gathered}
\end{equation*}
\notag
$$
Here we have a negative quantity on the left-hand side, so if we replace it by $0$, then we obtain a stronger restriction on $r$ and $\lambda$. Reducing by $p^r$ we obtain
$$
\begin{equation*}
0\leqslant 2p^{\lambda +2}-2p-p^{\lambda}+1=(p^{\lambda}-1)(2p^2-1)+2p^2-2p.
\end{equation*}
\notag
$$
Taking the assumptions $\lambda\geqslant 0$ and $p>1$ into account, it is obvious that this inequality is valid.
Theorem 2.25 is proved. Proof of Corollary 2.26. The implications $1)\Rightarrow 2)\Rightarrow 3)$ are obvious.
$3)\Rightarrow 1)$. By Theorem 2.11 we have $p=q$ and $\lambda =\mu p^k$ for some $k\in \mathbb Z$ and $\varrho_{\alpha,\delta}=\varrho_{\beta,\theta}$. Hence $\delta =\theta $ and $\alpha =\beta $. The proof is complete. AcknowledgementsThe author is obliged to S. I. Bogataya and A. A. Tuzhilin for their strong support.
Citation in format AMSBIB
\Bibitem{Bog23} |
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