Local metric properties and regular stratifications of p-adic definable sets

TL;DR

This paper investigates the local geometric and measure-theoretic properties of p-adic definable sets, establishing the existence of local density, introducing distinguished tangent cones, and deriving a p-adic version of the Cauchy-Crofton formula.

Contribution

It introduces the notion of distinguished tangent cones in the p-adic setting and proves their stabilization, extending real geometric concepts to p-adic definable sets.

Findings

Local density of p-adic definable sets exists at each point.

Distinguished tangent cones can be used to compute local density up to multiplicities.

A p-adic analogue of the Cauchy-Crofton formula for density is established.

Abstract

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone.We also prove that these distinguished tangent cones stabilize for small enough subgroups. We finally obtain the $p$-adic counterpart of the Cauchy-Crofton formula for the density. To prove these results we use the Lipschitz decomposition of definable $p$-adic sets of arXiv:0904.3853v1 and prove here the genericity of the regularity conditions for stratification such as $(w_f)$, $(w)$, $(a_f)$, $(b)$ and $(a)$ conditions.

Equations

$${\Theta }_d(X_a)=\underset{r\to 0}{\lim }\frac{{\mu }_d(X\cap B(a,r))}{{\mu }_d(B^d(0,r))},$$

$$r\mapsto \frac{{\mu }_d(X\cap B(a,r))}{r^d}$$

$$x\mapsto \{\begin{array}{ll}f(x)& \text{mbox}ifx\in R^m\\ 0& \text{mbox}else.\end{array}$$

$${\mu }_d(W):={\int }_{p(X)}{p}_{!}(1_W){\mu }_d$$

$$\{t\in K\mid |\alpha |{\square }_1|t-c|{\square }_2|\beta |,t-c\in \lambda P_n\},$$

$$\begin{array}[]{ll}\{(x,t)\in K^{m+1}\mid&x\in D,\ |\alpha(x)|{\operatorname{\square}\nolimits}_{1}|t-c(x)|{\operatorname{\square}\nolimits}_{2}|\beta(x)|,\\ &t-c(x)\in\lambda P_{n}\},\end{array}$$

$$|f_j(x,t)|=|h_{ij}(x)|\cdot |(t-c_i(x){)}^{a_{ij}}{\lambda }_i^{-a_{ij}}{|}^{\frac{1}{n_i}},\text{mbox}foreach(x,t)\in A_i,$$

$$|(t-c_i(x){)}^a{\lambda }_i^{-a}{|}^{\frac{1}{n_i}}\mathrm{ord}(t-c_i(x){)}^{s}h(x),$$

$$\frac{1}{k{r}^k}(\partial g_1/\partial r,\dots ,\partial g_1/\partial r{)}_{|r};$$

$$|f(x)-f(y)|\le \epsilon |x-y|.$$

$$X=\bigcup _{1\ge i\ge N(\epsilon )}{\gamma }_i({\Gamma }_i(\epsilon ))\cup Y.$$

$$(i_1,\dots ,i_m)=(1,\dots ,1,0,\dots ,0).$$

$$(x_1,\dots ,x_d)\mapsto (x_1,\dots ,x_{i-1},\frac{f_j(x_1,\dots ,x_d)}{\epsilon },x_{i+1},\dots ,x_d)$$

$${\mu }_1(X\cap B(0,2k))=(1-q^{-1})(q^{-2k}+q^{-2k-2}+\cdots )=\frac{q^{-2k}}{1+q^{-1}}.$$

$${\mu }_1(X\cap B(0,2k-1))=\frac{q^{-2k}}{1+q^{-1}}.$$

$$\underset{k\to \infty }{\lim }{q}^{2k}{\mu }_1(X\cap B(0,2k))=(1+q^{-1}{)}^{-1}$$

$$\underset{k\to \infty }{\lim }{q}^{2k-1}{\mu }_1(X\cap B(0,2k-1))=(1+q{)}^{-1}.$$

$$\underset{\genfrac{}{}{0ex}{}{n\to \infty }{n\equiv c\mathrm{mod}e}}{\lim }h(n)$$

$${\mathrm{MV}}_{\infty }(h):=\frac{1}{e}\sum _{c=0}^{e-1}\underset{\genfrac{}{}{0ex}{}{n\to \infty }{n\equiv c\mathrm{mod}e}}{\lim }h(n).$$

$${\mathrm{MV}}_{\infty }({\phi }_x)=\frac{1}{e}\sum _{c=0}^{e-1}{d}_c(x).$$

$$\phi (x,n)=\sum _{i=1}^k{n}^{{\ell }_i}{q}^{a_{i}n}{h}_{ic}(x)$$

$$\phi (x,n)=h_{jc}(x)$$

$$g_x:\mathbf{N}\to \mathbf{Q}:n\mapsto \phi (x,n)-{\phi }^{\prime }(x,n)$$

$$\sum _{i\in I}{n}^{{\ell }_i}{q}^{a_{i}n}{h}_{ic}(x)$$

$$g(x,n)=h_{jc}(x)$$

$$\gamma (\phi )(x,n):={\int }_{S(x,n)\cap X}\phi (y){\mu }_d,$$

$${\theta }_d(\phi )(x,n):=\frac{\gamma (\phi )(x,n)}{{\mu }_d(S_d(n))},$$

$${\Theta }_d(\phi ):={\mathrm{MV}}_{\infty }{\theta }_d(\phi ),$$

$${\Theta }_d(X)(x):={\Theta }_d(1_X)(x)$$

$${\Theta }_d(\phi )={\mathrm{MV}}_{\infty }({\theta }_d^{\prime }(\phi )),$$

Full text

Introduction

The present paper is devoted to the study of local metric properties of definable subsets of the $p$-adic affine space, with special stress on the local density of these subsets. It contains also results on tangent cones and existence of regular stratifications.

We shall start by recalling what is known in the real and complex context regarding the local density of (sub-)analytic sets. When $X_{a}$ is a germ at $a$ of a complex analytic subset $X$ of real dimension $d$ of the affine space ${\mathbf{C}}^{n}$, the local density $\Theta_{d}(X_{a})$ of $X_{a}$, sometimes called in this setting the Lelong number of $X_{a}$, has been introduced by Lelong in [30] as the limit of volumes of the intersection of a representative $X$ of the germ with suitably renormalized balls around $a$, namely

[TABLE]

where $B^{d}(0,r)$ is the real $d$-dimensional ball of centre [math] and radius $r>0$ and $\mu_{d}$ stands for $d$-dimensional volume. Lelong actually proved that the function

[TABLE]

decreases as $r$ goes to [math], pointing out, long before this concept has been formalized, the tame behaviour of the local normalized volume of analytic sets. Ten years after Lelong’s pioneering paper, Thie proved in [38] that the local density of a complex analytic subset $X$ at a point $a$ is a positive integer by expressing it as a sum of local densities of the components, counted with multiplicities, of the tangent cone of $X$ at $a$. Finally, more than twenty years after Lelong’s definition, Draper proved [17] that the local density is the algebraic multiplicity of the local ring of $X$ at $a$. The definition of local densities has been extended by Kurdyka and Raby to real subanalytic subsets of ${\mathbf{R}}^{n}$ in [29] (see also [28]). In fact, although the arguments in [29] and [28] which prove the existence of the density in the real subanalytic case were given before the notion of definable sets in o-minimal structures emerged, they apply to the real o-minimal setting. A short proof of this existence result, again produced just before the concept of tame definable sets and involving the Cauchy-Crofton formula, may be found in the seminal paper [31] for semi-pfaffian sets. In [11] (Theorem 1.3), one can find a proof in the real o-minimal setting of the existence of the local density viewed as the higher term of a finite sequence of localized curvature invariants involving the Cauchy-Crofton formula and the theory of regular stratifications. Of course, in the real setting, the density is in general no longer a positive integer, but a non negative real number, and Kurdyka and Raby proved an appropriate extension of Thie’s result by expressing again the local density in terms of the density of some components of the real tangent cone. The existence of the local density at each point of the closure of a subanalytic set is a manifestation of tameness of these sets near their singular points. Similarly, tameness in subanalytic geometry is also illustrated by the tame behavior of local density, viewed as a function of the base-point of the germ at which it is computed: it is actually proved in [12] that this function along a given global subanalytic set is a Log-analytic function, that is, a polynomial in subanalytic functions and their logarithms (see Siu’s paper [37] for similar results in the complex case). Draper’s result has been extended to the real setting by Comte in [10] in the following way. Recall that if $X$ is a complex analytic subset of the affine space of complex dimension $d$ at $x$, the algebraic multiplicity of the local ring of $X$ at $x$ is equal to the local degree of a generic linear projection $p:X\rightarrow{\mathbf{C}}^{d}$, that is, to the number of points near $x$ in a generic fiber of $p$. Over the reals, if $X$ is of local dimension $d$ at $x$, the number of points in fibers (near $x$) of a generic linear projection $p:X\rightarrow{\mathbf{R}}^{d}$ is not generically constant near $p(x)$ in general. The idea introduced in [10] to overcome this difficulty is to consider as a substitute for the local degree of $p$ the sum $d(p):=\sum_{i\in{\mathbf{N}}}i\cdot\theta_{i}$, where $\theta_{i}$ is the local density at $p(x)$ of the germ of the set of points in ${\mathbf{R}}^{d}$ over which the fiber of $p$ has exactly $i$ points near $x$. The so-called local Cauchy-Crofton formula proved in [10] states that the average along all linear projections $p$ of the degrees $d(p)$ is equal to the local density of $X$ at $x$ and can be considered as the real analogue of Draper’s result. Finally, the complete multi-dimensional version of the local Cauchy-Crofton formula for real subanalytic sets is presented in [11] (Theorem 3.1), where the multi-dimensional substitute of the [math]-dimensional local degree $d(p)$ is obtained by considering the local Euler characteristic of generic multi-dimensional fibers, instead of the local number of points.

Now let $K$ be a finite extension of ${\mathbf{Q}}_{p}$ and $X$ be a definable subset (semi-algebraic or subanalytic) of $K^{n}$. Let $x$ be a point of $K^{n}$. When one tries to define the local density of $X$ at $x$ similarly to the archimedean case, one is faced to the problem, illustrated in 2.1, that the limit of local volumes in general no longer exists. It appears that the normalized volumes $v_{n}$ of $X$ in the balls $B(x,n):=\{w\in K\mid{\rm ord}(x-w)\geq n\}$ has a periodic convergence, that is to say, there exists an integer $e>0$, such that for all $c=0,\cdots,e-1$, $(v_{c+m\cdot e})_{m\in{\mathbf{N}}}$ has a limit $v_{c}$ in ${\mathbf{Q}}$ (see Proposition 2.2.3), with possibly $v_{c}\not=v_{c^{\prime}}$, for $c\not=c^{\prime}$. The reader having essentially in mind the real case is thus strongly encouraged to start reading this article by the example studied in 2.1 that emphasizes this phenomenon. We resolve that issue by using an appropriate renormalization device that leads us to express the mean value $\displaystyle\frac{1}{e}\cdot\sum_{c=0}^{e-1}v_{c}$ as the local density of $X$ at $x$.

Another new issue occurring in the $p$-adic setting is the lack of a natural notion of a tangent cone. Unlike the real case where only the action of the multiplicative group ${\mathbf{R}}_{+}^{\times}$ has to be considered, in the $p$-adic case, there seems to be no preferential subgroup of $K^{\times}$ at hand. We remedy this by introducing, for each definable open subgroup of finite index $\Lambda$ in $K^{\times}$, a tangent cone $C_{x}^{\Lambda}$ at $x$ which is stable by homotheties in $\Lambda$, that is, which is a $\Lambda$-cone. One should note that such $\Lambda$-cones were already considered more than twenty years ago in the work of Heifetz on $p$-adic oscillatory integrals and wave front sets [22]. Nevertheless, we prove in Theorem 5.6.1 that, given a definable subset $X$, among these cones, some are distinguished as maximal for an inclusion property, and appear as the good tangent cones to be considered, in the sense that they capture the local geometry of our set. We are then able, by deformation to the tangent cone, to assign multiplicities to points in the tangent cone $C_{x}^{\Lambda}$.

Our main results regarding $p$-adic local densities are Theorem 3.6.2, which is a $p$-adic analogue of the result of Thie and Kurdyka-Raby, and Theorem 6.2.1, which is a $p$-adic analogue of Comte’s local Cauchy-Crofton formula. An important technical tool in our proof of Theorem 3.6.2 is provided by Theorem 5.3.6 which allows us to decompose our definable set into Lipschitz graphs. Such a regular decomposition has been obtained in [5] and extends to the $p$-adic setting a real subanalytic result of [26]. In section 4 we prove the existence of $(w_{f})$-regular stratifications for definable $p$-adic functions, and consequently the existence of Thom’s $(a_{f})$-regular stratifications for definable $p$-adic functions, $(w)$-regular, or Verdier regular, stratifications, and Whitney’s $(b)$-regular stratifications for $p$-adic definable sets (Theorem 4.2.5).

During the preparation of this paper, the authors have been partially supported by grant ANR-06-BLAN-0183. We also thank the Fields Institute of Toronto, where this paper was partly written, for bringing us exceptional working conditions.

Contents

1. 1 Preliminaries

2. 1.1 Definable sets over the $p$-adics

3. 1.2 The $p$-adic measure

4. 1.3 Adding sorts

5. 1.4 $p$-adic Cell Decomposition

6. 1.5

7. 2 Local densities

8. 2.1 A false start

9. 2.2 Mean value at infinity of bounded constructible functions

10. 2.3 Local densities

11. 2.4

12. 3 Tangent cones

13. 3.1 Cones

14. 3.2 Local conic structure of definable sets

15. 3.3 Tangent cones

16. 3.4 More on $\varepsilon$-analytic functions

17. 3.5 Deformation to the tangent cone

18. 3.6 Multiplicities on the tangent cones

19. 4 Existence of $(w_{f})$-regular stratifications

20. 4.1

21. 4.2

22. 4.3

23. 5 Proof of Theorem 3.6.2 and existence of distinguished tangent $\Lambda$-cones

24. 5.1 Proof of Theorem 3.6.2: a first reduction

25. 5.2 Proof of Theorem 3.6.2: the case $d=n$

26. 5.3 Graphs

27. 5.4 An alternative view on cones with multiplicities.

28. 5.5 End of proof of Theorem 3.6.2

29. 5.6 Existence of distinguished tangent $\Lambda$-cones

30. 6 A local Crofton formula

31. 6.1 Local direct image

32. 6.2 The local Crofton formula for the local density

1 Preliminaries

1.1 Definable sets over the $p$-adics

Let $K$ be a finite field extension of ${\mathbf{Q}}_{p}$ with valuation ring $R$. We denote by ${\rm ord}$ the valuation and set $|x|:=q^{-{\rm ord}(x)}$ and $|0|=0$, with $q$ cardinality of the residue field of $K$. If $x=(x_{i})$ is a point in $K^{m}$ and $n$ is an integer, we denote by $B(x,n)$ the ball in $K^{m}$ given by the conditions ${\rm ord}(z_{i}-x_{i})\geq n$, $1\leq i\leq m$.

We recall the notion of (globally) subanalytic subsets of $K^{n}$ and of semi-algebraic subsets of $K^{n}$. Let ${\mathcal{L}}_{\rm Mac}=\{0,1,+,-,\cdot,\{P_{n}\}_{n>0}\}$ be the language of Macintyre and ${\mathcal{L}}_{\rm an}={\mathcal{L}}_{\rm Mac}\cup\{^{-1},\cup_{m>0}K\{x_{1},\ldots,x_{m}\}\}$, where $P_{n}$ stands for the set of $n$th powers in $K^{\times}$, where -1 stands for the field inverse extended on [math] by $0^{-1}=0$, where $K\{x_{1},\ldots,x_{m}\}$ is the ring of restricted power series over $K$ (that is, formal power series converging on $R^{m}$), and each element $f$ of $K\{x_{1},\ldots,x_{m}\}$ is interpreted as the restricted analytic function $K^{m}\to K$ given by

[TABLE]

By subanalytic we mean ${\mathcal{L}}_{\rm an}$-definable with coefficients from $K$ and by semi-algebraic we mean ${\mathcal{L}}_{\rm Mac}$-definable with coefficients from $K$. Note that subanalytic, resp. semi-algebraic, sets can be given by a quantifier free formula with coefficients from $K$ in the language ${\mathcal{L}}_{\rm Mac}$, resp. ${\mathcal{L}}_{\rm an}$.

In this section we let ${\mathcal{L}}$ be either the language ${\mathcal{L}}_{\rm Mac}$ or ${\mathcal{L}}_{\rm an}$ and by ${\mathcal{L}}$-definable we will mean semi-algebraic, resp. subanalytic when ${\mathcal{L}}$ is ${\mathcal{L}}_{\rm Mac}$, resp. ${\mathcal{L}}_{\rm an}$. Everything in this paper will hold for both languages and we will give appropriate references for both languages when needed.

For each definable set $X\subset K^{n}$, let ${\mathcal{C}}(X)$ be the ${\mathbf{Q}}$-algebra of functions on $X$ generated by functions $|f|$ and ${\rm ord}(f)$ for all definable functions $f:X\to K^{\times}$.

We refer to [19] and [16] for the definition of the dimension of ${\mathcal{L}}$-definable sets.

1.2 The $p$-adic measure

Suppose that $X\subset K^{n}$ is an ${\mathcal{L}}$-definable set of dimension $d\geq 0$. The set $X$ contains a definable nonempty open $K$-analytic submanifold $X^{\prime}\subset K^{n}$ such that $X\setminus X^{\prime}$ has dimension $<d$, cf. [16]. There is a canonical $d$-dimensional measure $\mu_{d}$ on $X^{\prime}$ coming from the embedding in $K^{n}$, which is constructed as follows, cf. [36]. For each $d$-element subset $J$ of $\{1,\ldots,n\}$, with $j_{i}<j_{i+1}$, $j_{i}$ in $J$, let $dx_{J}$ be the $d$-form $dx_{j_{1}}\wedge\ldots\wedge dx_{j_{d}}$ on $K^{n}$, with $x=(x_{1},\ldots,x_{n})$ standard global coordinates on $K^{n}$. Let $x_{0}$ be a point on $X^{\prime}$ such that $x_{I}$ are local coordinates around $x_{0}$ for some $I\subset\{1,\ldots,n\}$. For each $d$-element subset $J$ of $\{1,\ldots,n\}$ let $g_{J}$ be the ${\mathcal{L}}$-definable function determined at a neigborhood of $x_{0}$ in $X^{\prime}$ by $g_{J}dx_{I}=dx_{J}$. There is a unique volume form $|\omega_{0}|_{X^{\prime}}$ on $X^{\prime}$ which is locally equal to $(\max_{J}|g_{J}|)|dx_{I}|$ around every point $x_{0}$ in $X^{\prime}$. Indeed, $|\omega_{0}|_{X^{\prime}}$ is equal to $\sup_{J}|dx_{J}|$. The canonical $d$-dimensional measure $\mu_{d}$ on $X^{\prime}$ (cf. [36] [34]), is the one induced by the volume form $|\omega_{0}|_{X^{\prime}}$. We extend this measure to $X$ by zero on $X\setminus X^{\prime}$ and still denote it by $\mu_{d}$.

1.3 Adding sorts

By analogy with the motivic framework, we now expand the language ${\mathcal{L}}$ to a three sorted language ${\mathcal{L}}^{\prime}$ having ${\mathcal{L}}$ as language for the valued field sort, the ring language ${\mathbf{L}}_{\rm Rings}$ for the residue field, and the Presburger language ${\mathbf{L}}_{\rm PR}$ for the value group together with maps ${\rm ord}$ and ${\overline{\rm ac}}$ as in [8]. By taking the product of the measure $\mu_{m}$ with the counting measure on $k_{K}^{n}\times{\mathbf{Z}}^{r}$ one defines a measure still denoted by $\mu_{m}$ on $K^{m}\times k_{K}^{n}\times{\mathbf{Z}}^{r}$.

One defines the dimension of an ${\mathcal{L}}^{\prime}$-definable subset $X$ of $K^{m}\times k_{K}^{n}\times{\mathbf{Z}}^{r}$ as the dimension of its projection $p(X)\subset K^{m}$. If $X$ is of dimension $d$, one defines a measure $\mu_{d}$ on $X$ extending the previous construction on $X$ by setting

[TABLE]

with $p_{!}({\mathbf{1}}_{W})$ the function $y\mapsto{\rm card}(p^{-1}(y)\cap W)$.

For such an $X$, one defines ${\mathcal{C}}(X)$ as the ${\mathbf{Q}}$-algebra of functions on $X$ generated by functions $\alpha$ and $q^{-\alpha}$ with $\alpha:X\rightarrow{\mathbf{Z}}$ definable in ${\mathcal{L}}^{\prime}$. Note that this definition coincides with the previous one when $n=r=0$. Since ${\mathcal{L}}^{\prime}$ is interpretable in ${\mathcal{L}}$, the formalism developed in this section extends to ${\mathcal{L}}^{\prime}$-definable objects in a natural way.

1.4 $p$-adic Cell Decomposition

Cells are defined by induction on the number of variables

1.4.1 Definition**.**

An ${\mathcal{L}}$-cell $A\subset K$ is a (nonempty) set of the form

[TABLE]

with constants $n>0$, $\lambda,c$ in $K$, $\alpha,\beta$ in $K^{\times}$, and $\square_{i}$ either $<$ or no condition. An ${\mathcal{L}}$-cell $A\subset K^{m+1}$, $m\geq 0$, is a set of the form

[TABLE]

with $(x,t)=(x_{1},\ldots,x_{m},t)$, $n>0$, $\lambda$ in $K$, $D=p_{m}(A)$ a cell where $p_{m}$ is the projection $K^{m+1}\to K^{m}$, ${\mathcal{L}}$-definable functions $\alpha,\beta:K^{m}\to K^{\times}$ and $c:K^{m}\to K$, and $\square_{i}$ either $<$ or no condition, such that the functions $\alpha,\beta$, and $c$ are analytic on $D$. We call $c$ the center of the cell $A$ and $\lambda P_{n}$ the coset of $A$. In either case, if $\lambda=0$ we call $A$ a [math]-cell and if $\lambda\not=0$ we call $A$ a $1$-cell.

In the $p$-adic semi-algebraic case, Cell Decomposition Theorems are due to Cohen [9] and Denef [13], [15] and they were extended in [3] to the subanalytic setting where one can find the following version.

1.4.2 Theorem** ($p$-adic Cell Decomposition).**

Let $X\subset K^{m+1}$ and $f_{j}:X\to K$ be ${\mathcal{L}}$-definable for $j=1,\ldots,r$. Then there exists a finite partition of $X$ into ${\mathcal{L}}$-cells $A_{i}$ with center $c_{i}$ and coset $\lambda_{i}P_{n_{i}}$ such that

[TABLE]

with $(x,t)=(x_{1},\ldots,x_{m},t)$, integers $a_{ij}$, and $h_{ij}:K^{m}\to K$ ${\mathcal{L}}$-definable functions which are analytic on $p_{m}(A_{i})$, $j=1,\ldots,r$. If $\lambda_{i}=0$, we use the convention that $a_{ij}=0$.

Let us also recall the following lemma from [4].

1.4.3 Lemma**.**

Let $X\subset K^{m+1}$ be ${\mathcal{L}}$-definable and let $G_{j}$ be functions in ${\mathcal{C}}(X)$ in the variables $(x_{1},\ldots,x_{m},t)$ for $j=1,\ldots,r$. Then there exists a finite partition of $X$ into ${\mathcal{L}}$-cells $A_{i}$ with center $c_{i}$ and coset $\lambda_{i}P_{n_{i}}$ such that each restriction $G_{j}|_{A_{i}}$ is a finite sum of functions of the form

[TABLE]

where $h$ is in ${\mathcal{C}}(K^{m})$, and $s\geq 0$ and $a$ are integers.

The following $p$-adic curve selection lemma is due to van den Dries and Scowcroft [19] in the semi-algebraic case and to Denef and van den Dries [16] in the subanalytic case. The statement is the $p$-adic counterpart of the semi-algebraic or subanalytic curve selection lemma over the reals.

1.4.4 Lemma** (Curve Selection).**

Let $A$ be a definable subset of $K^{n}$ and let $x$ be in $\overline{A}$. Then there exists a definable function $f=(f_{1},\ldots,f_{n}):R\to K^{n}$ such that the $f_{i}$ are given by power series (over $K$) converging on $R$, such that $f(0)=x$, and such that $f(R\setminus\{0\})\subset A$.

The following is a $p$-adic analogue of a classical lemma by Whitney (see [41]).

1.4.5 Lemma** ($p$-adic Whitney Lemma).**

Let $g:R\to K^{n}$ be a map given by $n$ analytic power series over $K$, converging on $R$, such that the map $g$ is nonconstant. Then, the limit $\ell\in{\mathbf{P}}^{n-1}(K)$ for $r\to 0$ of the lines $\ell_{r}\in{\mathbf{P}}^{n-1}(K)$ connecting $g(0)$ with $g(r)$ exists. Also the limit $\ell^{\prime}$ of the tangent lines $\ell_{r}^{\prime}:=\{g(r)+\lambda(\partial g_{1}/\partial r,\ldots,\partial g_{1}/\partial r)_{|r}\mid\lambda\in K\}$ for $r\to 0$ exists and $\ell^{\prime}=\ell$.

Proof.

Since $g$ is nonconstant, for $r\not=0$ close to [math] one has $g(r)\not=g(0)$ and $(\partial g_{1}/\partial r,\ldots,\partial g_{1}/\partial r)_{|r}\not=0$, and hence, $\ell_{r}$ and $\ell^{\prime}_{r}$ are well-defined for $r\not=0$ close to [math]. We may suppose that $g(0)=0$ and that each of the $g_{i}$ is nonconstant. Write $g_{i}(r)=\sum_{j\geq 0}a_{ij}r^{j}$ with $a_{ij}\in K$ and for each $i$, let $k_{i}$ be the smallest index $j$ such that $a_{ij}\not=0$. Then $k_{i}>0$ for each $i$ since $g(0)=0$. Let $k$ be the minimum of the $k_{i}$. Then clearly $\ell$ and $\ell^{\prime}$ are the same line $\ell$ connecting [math] and $(a_{1k},\ldots,a_{nk})\not=0$. Indeed, the line $\ell_{r}$ connects [math] and $g(r)$ which is equivalent to connecting [math] and $g(r)/r^{k}$; the point $g(r)/r^{k}$ converges to $(a_{1k},\ldots,a_{nk})$ and thus $\ell_{r}$ converges to the line $\ell$. Likewise, the line $\ell^{\prime}_{r}$ connects [math] and $(\partial g_{1}/\partial r,\ldots,\partial g_{1}/\partial r)_{|r}$ which is equivalent to connecting [math] and

[TABLE]

the point $\frac{1}{kr^{k}}(\partial g_{1}/\partial r,\ldots,\partial g_{1}/\partial r)_{|r}$ converges to $(a_{1k},\ldots,a_{nk})$ and thus also $\ell_{r}^{\prime}$ converges to $\ell$ when $r\to 0$. ∎

1.5

Fix two integers $d\leq m$. Let $U$ be an open definable subset of $K^{d}$ and let $\varphi$ be a definable analytic mapping $U\rightarrow K^{m-d}$. We view the graph $\Gamma(\varphi)$ of $\varphi$ as a definable subset of $K^{m}$. Let $\varepsilon$ be a positive real number. We say that $\varphi$ is $\varepsilon$-analytic, if the norm $|D\varphi|=\max_{i,j}|\partial\varphi_{i}/\partial x_{j}|$ of the differential of $\varphi$ is less or equal than $\varepsilon$ at every point of $U$.

For $\varepsilon>0$, call a function $f:D\to K^{m}$ on a subset $D$ of $K^{n}$ $\varepsilon$-Lipschitz when for all $x,y\in D$ one has

[TABLE]

The function $f$ is called locally $\varepsilon$-Lipschitz when for each $x\in D$ there exists an open subset $U$ of $K^{n}$ containing $x$ such that the restriction of $f$ to $U\cap D$ is $\varepsilon$-Lipschitz.

1.5.1 Lemma**.**

Let $U$ be open in $K^{n}$ and $f:U\to K^{m}$ a function which is $\varepsilon$-analytic. Then $f$ is locally $\varepsilon$-Lipschitz.

Proof.

Choose $u\in U$, and a basic neighborhood $U_{u}$ of $u$ in $U$ such that the component functions $f_{i}$ of $f$ are given by converging power series on $U_{u}$, where basic neighborhood means a ball of the form $c+\lambda R^{n}$ with $c\in K^{n}$ and $\lambda\in K^{\times}$. We may suppose that $U_{u}=R^{n}$, that $u=0$, and that $\varepsilon=1$. We may also assume that for each $i,j$, the partial derivative $\partial f_{j}(x)/\partial x_{i}$ is bounded in norm by $1$ on $U_{u}$. Since $|\partial f_{j}(x)/\partial x_{i}(0)|\leq 1$, it follows that the linear term of $f_{j}$ in $x_{i}$ has a coefficient of norm $\leq 1$ for each $i,j$. By the convergence of the power series, the coefficients of the $f_{j}$ are bounded in norm, say by $N$, and we can put $U^{\prime}:=\{x\in R^{n}\mid|x|<1/N\}$. Clearly $U^{\prime}$ contains $u=0$. By the non-archimedean property of the $p$-adic valuation, the restriction of $f$ to $U^{\prime}$ is $1$-Lipschitz. ∎

For more results related to Lipschitz continuity on the $p$-adics, see [5] or Theorem 5.3.6 below. The following lemma is a partial converse of Lemma 1.5.1, especially in view of the fact that any definable function is piecewise analytic.

1.5.2 Lemma**.**

Let $U$ be a definable open in $K^{n}$ and let $f:D\to K^{m}$ be a definable analytic function which is locally $\varepsilon$-Lipschitz. Then $f$ is $\varepsilon$-analytic.

Proof.

We proceed by contradiction. Suppose that $|Df|>\varepsilon$ at $u\in U$. Choose a basic neighborhood $U_{u}$ of $u$ in $U$ such that the component functions $f_{i}$ of $f$ are given by converging power series on $U_{u}$, (here again by basic neighborhood we mean a ball of the form $c+\lambda R^{n}$ with $c\in K^{n}$ and $\lambda\in K^{\times}$). We may suppose that $U_{u}=R^{n}$, that $u=0$, and that $\varepsilon=1$. By assumption, we have for some $i,j$ that $|(\partial f_{j}(x)/\partial x_{i})(0)|>1$, hence, the linear term of $f_{j}$ in $x_{i}$ has a coefficient of norm strictly greater than $1$. By the convergence of the power series, the coefficients of the $f_{j}$ are bounded in norm, say by $N$, and thus for any $x$ in $\{x\in R^{n}\mid|x|<1/N\}$ one has $|f(0)-f(x)|>|x|$ which contradicts the fact that $f$ is $\varepsilon$-Lipschitz with $\varepsilon=1$. ∎

The following is the $p$-adic analogue of Proposition 1.4 of [29].

1.5.3 Proposition**.**

Let $X$ be a definable subset of dimension $d$ of $K^{m}$. For every $\varepsilon>0$, there exists a definable subset $Y$ of $X$ of dimension $<d$, $N(\varepsilon)\geq 0$, definable open subsets $U_{i}(\varepsilon)$ of $K^{d}$, for $1\leq i\leq N(\varepsilon)$, definable, $\varepsilon$-analytic functions $\varphi_{i}(\varepsilon):U_{i}(\varepsilon)\rightarrow K^{m-d}$, with graphs $\Gamma_{i}(\varepsilon)$, and elements $\gamma_{1}$, …, $\gamma_{N(\varepsilon)}$ in ${\rm GL}_{m}(R)$ such that the sets $\gamma_{i}(\Gamma_{i}(\varepsilon))$ are all disjoint and contained in $X$, and

[TABLE]

Proof.

Take a finite partition of $X$ into cells $X_{i}$. By neglecting cells of dimension $<d$, we may suppose that $X=X_{1}$ and that $X_{1}$ is a cell of type $(i_{1},\ldots,i_{m})$ with $\sum_{j=1}^{m}i_{j}=d$. By reordering the coordinates, by partitioning again into cells, and neglecting cells of dimension $<d$, we may suppose that $X$ is a cell of type

[TABLE]

Let $p:K^{m}\to K^{d}$ be the projection to the first $d$ coordinates.

The set $X$ itself is already a graph of a map $(f_{1},\ldots,f_{m-d})=f:p(X)\to K^{m-d}$. We may suppose that the $f_{i}$ are analytic. Write $(x_{1},\ldots,x_{d})$ for the coordinates on $p(X)$. Consider the differentials $(\frac{\partial f_{j}}{\partial x_{i}})_{i=1}^{d}$. By partitioning further, we may suppose that for each $i=1,\ldots,d$, either $|\frac{\partial f_{j}}{\partial x_{i}}|\leq\varepsilon$ or $|\frac{\partial f_{j}}{\partial x_{i}}|>\varepsilon$ hold on $p(X)$. If $|\frac{\partial f_{j}}{\partial x_{i}}|>\varepsilon$ on $p(X)$ for some $i$ and $j$, then we may suppose that

[TABLE]

is a bi-analytic change of variables (cf. Corollary 3.7 of [5]). By performing such change of variables successively, we are done by the chain rule for differentiation. ∎

The following lemma is classical, see, for example, [18] for the semi-algebraic case and, for example, [6] for the subanalytic case. Let ${\mathcal{L}}^{\ast}$ be the language ${\mathcal{L}}$ together with a function symbol for the field inverse on $K^{\times}$ (extended by zero on zero), function symbols for each $n$ which stands for a (definable) $n$-th root picking function $\sqrt[n]{}$ on the $n$-th powers (extended by zero outside the $n$-th powers), and for each degree $n$ a Henselian root picking function $h_{n}$ for polynomials of degree $n$ in the $n+1$ coefficients (extended by zero if the conditions of Hensel’s Lemma are not fulfilled).

1.5.4 Lemma** ([18], [6]).**

Let $f:D\subset K^{n}\to K^{m}$ be an ${\mathcal{L}}$-definable function. Then $D$ can be partitioned into finitely many definable pieces $D_{i}$ such that there are ${\mathcal{L}}^{\ast}$-terms $t_{i}$ with $f(x)=t_{i}(x)$ for each $i$ and each $x\in D_{i}$.

2 Local densities

2.1 A false start

Let $X$ be a definable subset of $K^{m}$ of dimension $d$ and let $x$ be a point of $K^{m}$. Considering what is already known in the complex analytic and real o-minimal case, a natural way to define the local density of $X$ at $x=(x_{1},\dots,x_{m})$ would be to consider the limit of $q^{nd}\mu_{d}(X\cap B(x,n))$, as $n\to\infty$. Unfortunately this naïve attempt fails as is shown by the following example that we present in detail in order to caution the reader not to rely too heavily on intuition coming from the real setting. Take $X$ the subset of points of even valuation in $K$ and $x=0$. Write $\pi_{K}$ for a uniformizer of $R$. The unit ball $B$ in $K$ being of measure $1$, the ball $\pi_{K}^{\ell}\cdot B=B(0,\ell)$ of radius $q^{-\ell}$ has volume $q^{-\ell}$ and, by consequence, the sphere $\pi_{K}^{\ell}\cdot S=\pi_{K}^{\ell}\cdot B\setminus\pi_{K}^{\ell+1}\cdot B$ of radius $q^{-\ell}$ has volume $q^{-\ell}(1-q^{-1})$. For $k\in{\mathbf{N}}$, let us first compute the volume of $X\cap B(0,2k)$. The set $X\cap B(0,2k)$ is the disjoint union of the spheres $\pi_{K}^{2j}\cdot S$ for $j\geq k$ and thus has as volume

[TABLE]

On the other hand, the set $X\cap B(0,2k-1)$ is also the disjoint union of the spheres $q^{2j}\cdot S$ for $j\geq k$ and thus has as volume

[TABLE]

We finally see that in this example the value of the limit $\displaystyle\lim_{\ell\to\infty}\frac{\mu_{1}(X\cap B(0,\ell))}{\mu_{1}(B(0,\ell))}$ depends on the parity of $\ell$, since

[TABLE]

and

[TABLE]

In our example one notices that the convergence of the ratio $\displaystyle\frac{\mu_{1}(X\cap B(0,\ell))}{\mu_{1}(B(0,\ell))}$ is $2$-periodic and that one may recover the expected local density, which should be $\displaystyle\frac{1}{2}$, by taking the average of the two limits. To obviate the kind of difficulty presented by this example (the periodic convergence), we are led to introduce a regularization device that we shall explain now.

2.2 Mean value at infinity of bounded constructible functions

We will use the following elementary definition of the mean value at infinity of certain real valued functions on ${\mathbf{N}}$.

2.2.1 Definition**.**

Say that a function $h:{\mathbf{N}}\to{\mathbf{R}}$ has a mean value at infinity if there exists an integer $e>0$ such that

[TABLE]

exists in ${\mathbf{R}}$ for each $c=0,\ldots,e-1$ and in this case define the mean value at infinity of $h$ as the average

[TABLE]

Clearly the value ${\rm MV}_{\infty}(h)$ is independent of the choice of the modulus $e>0$.

Let $X$ be a definable subset of $K^{m}$, so that $X\times{\mathbf{N}}$ is a definable subset of $K^{m}\times{\mathbf{Z}}$. Say that a real valued function $g$ on $X\times{\mathbf{N}}$ is $X$-bounded if for every $x$ in $X$ the restriction of $g$ to $\{x\}\times{\mathbf{N}}$ is bounded (in the sense that $g(\{x\}\times{\mathbf{N}})$ is contained in a compact subset of ${\mathbf{R}}$). As has been indicated in the introduction and in the example of section 2.1, for an $X$-bounded function $\varphi$ in ${\mathcal{C}}(X\times{\mathbf{N}})$ and $x\in X$, the function $\varphi_{x}:{\mathbf{N}}\to{\mathbf{Q}}:n\mapsto\varphi(x,n)$ may not have a unique limit for $n\to\infty$, but it may have a mean value at infinity ${\rm MV}_{\infty}(\varphi_{x})$, as we will indeed show in Proposition 2.2.3. We will moreover show in Proposition 2.2.3 that ${\rm MV}_{\infty}(\varphi_{x})$, considered as a function in $x\in X$, lies in ${\mathcal{C}}(X)$ and that ${\rm MV}_{\infty}(\varphi_{x})$ can be calculated using a single integer $e$ as modulus when $x$ varies in $X$.

2.2.2 Lemma**.**

Let $\varphi$ be in in ${\mathcal{C}}(X\times{\mathbf{N}})$. Suppose that, for each $x\in X$, the function $\varphi_{x}:{\mathbf{N}}\to{\mathbf{Q}}:n\mapsto\varphi(x,n)$ has finite image. Then $\varphi_{x}$ has a mean value at infinity ${\rm MV}_{\infty}(\varphi_{x})$ for each $x$. Moreover, there exist a definable function $b:X\to{\mathbf{N}}$ and an integer $e>0$ such that for all $c$ with $0\leq c<e$ and all $x\in X$, the rational number $d_{c}(x):=\varphi(x,n)$ is independent of $n$ as long as $n\geq b(x)$ and $n\equiv c\bmod e$. Thus, for each $x\in X$, one has

[TABLE]

By consequence, the function ${\rm MV}_{\infty}(\varphi_{x})$, considered as a function in $x\in X$, lies in ${\mathcal{C}}(X)$.

Proof.

The lemma is a direct consequence of Lemma 1.4.3 and quantifier elimination in the three sorted language ${\mathcal{L}}^{\prime}$ of section 1.3. Indeed, for $\varphi\in{\mathcal{C}}(X\times{\mathbf{N}})$ there exist, by Lemma 1.4.3 and quantifier elimination in ${\mathcal{L}}^{\prime}$, a definable function $b:X\to{\mathbf{N}}$ and an integer $e>0$ such that for all $c$ with $0\leq c<e$ one has

[TABLE]

for all $x\in X$ and all $n$ with $n\geq b(x)$ and $n\equiv c\bmod e$, and where the $h_{ic}$ are in ${\mathcal{C}}(X)$. Clearly, by regrouping, we may suppose that the pairs $(\ell_{i},a_{i})$ are mutually different. But then, since $\varphi_{x}$ has finite image for each $x\in X$, one must find

[TABLE]

for all $x\in X$ and all $n$ with $n\geq b(x)$ and $n\equiv c\bmod e$, where $j$ is such that $(\ell_{j},a_{j})=(0,0)$. Hence, one has $h_{jc}=d_{c}$ and we are done. ∎

2.2.3 Proposition**.**

Let $\varphi$ be in ${\mathcal{C}}(X\times{\mathbf{N}})$. Suppose that $\varphi$ is $X$-bounded. Then there exist $\varphi^{\prime}$ in ${\mathcal{C}}(X\times{\mathbf{N}})$ with $\lim_{n\to\infty}\varphi^{\prime}(x,n)=0$ for all $x\in X$ and such that the function

[TABLE]

has finite image. Clearly, the function $g:X\times{\mathbf{N}}:(x,n)\mapsto g_{x}(n)$ lies in ${\mathcal{C}}(X\times{\mathbf{N}})$. Hence, ${\rm MV}_{\infty}(g_{x})$ and ${\rm MV}_{\infty}(\varphi_{x})$ exist and are equal and the function ${\rm MV}_{\infty}(\varphi_{x})$, considered as a function in $x\in X$, lies in ${\mathcal{C}}(X)$. Also, if $\varphi\geq 0$ then ${\rm MV}_{\infty}(\varphi_{x})\geq 0$ for all $x\in X$.

Proof.

Write again $\varphi$ as in (2.2.1) for some integer $e$, where again the pairs $(\ell_{i},a_{i})$ are mutually different. Define $\varphi^{\prime}(x,n)$ as the partial sum

[TABLE]

for $x\in X$ and $n$ satisfying $n\geq b(x)$ and $n\equiv c\bmod e$, where $c=0,\ldots,e-1$, where $I$ consists of those $i$ with $a_{i}<0$. Extend $\varphi^{\prime}$ to the whole of $X\times{\mathbf{N}}$ by putting it equal to $\varphi$ for those $n$ with $n<b(x)$. Since $\varphi$ is $X$-bounded, one must have that $a_{i}\leq 0$ for all $i$, and, for those $i$ with $a_{i}=0$ one must have $\ell_{i}=0$. But then, we find

[TABLE]

for all $x\in X$ and all $n$ with $n\geq b(x)$ and $n\equiv c\bmod e$, where $j$ is such that $(\ell_{j},a_{j})=(0,0)$. For $n$ with $n<b(x)$ one clearly has $g(x,n)=0$. The conclusions now follow from Lemma 2.2.3. ∎

2.3 Local densities

As already sketched in the introduction, we will define the local density of an ${\mathcal{L}}$-definable set $X\subset K^{m}$ at a point $x$ as the mean value at infinity of the renormalized measure of the intersection of $X$ with the sphere of radius $q^{-n}$ around $x$. At our disposal to show that this is well-defined we have Proposition 2.2.3 and Lemma 2.3.1 below which guarantee the existence of the mean value at infinity. More generally, for a bounded function $\varphi$ in ${\mathcal{C}}(X)$, we extend $\varphi$ to $K^{m}$ by zero outside $X$ and we will define the density of $\varphi$ at any point $x\in K^{m}$ by a similar procedure, replacing the measure by an integral of $\varphi$ on a small sphere around $x$.

Let $\varphi$ be a bounded function in ${\mathcal{C}}(X)$, meaning that the image of $\varphi$ is contained in a compact subset of ${\mathbf{R}}$. For $(x,n)$ in $K^{m}\times{\mathbf{N}}$ we set

[TABLE]

where $S(x,n)$ is the sphere $\{y\in K^{m}\mid|x-y|=q^{-n}\}$ of radius $q^{-n}$ around $x$. Note that, by Lemma 2.3.3 below, one could as well work with balls around $x$ instead of spheres consequently in this section. By [14] for the semi-algebraic case and [3] for the subanalytic case, the function $\gamma(\varphi):(x,n)\mapsto\gamma(\varphi)(x,n)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$.

Suppose that $X$ is of dimension $d$. Then we renormalize $\gamma(\varphi)$ by dividing it by the volume of the $d$-dimensional sphere of corresponding radius and define the resulting function $\theta_{d}(\varphi)$ by

[TABLE]

where $S_{d}(n)$ is the $d$-dimensional sphere of radius $q^{-n}$, namely the set $\{w\in K^{d}\mid|w|=q^{-n}\}$. Note that $S_{d}(n)$ has measure equal to $(1-q^{-d})q^{-nd}$ and thus, $\theta_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$.

The following lemma yields sufficient conditions for the mean value at infinity of $\theta_{d}(\varphi)$ to exist, in view of Proposition 2.2.3.

2.3.1 Lemma**.**

Let $\varphi$ be a bounded function in ${\mathcal{C}}(X)$. Assume $X$ is of dimension $d$. Then the function $\theta_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$ and is $K^{m}$-bounded.

Proof.

That $\theta_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$ is shown above, so we just have to show that $\theta_{d}(\varphi)$ is $K^{m}$-bounded. By the additivity of integrals and by cell decomposition, we may suppose that $X$ is a cell of dimension $d$. By changing the order of the coordinates if necessary and by Proposition 4.2.3, we may suppose that $X$ projects isometrically to the first $d$ coordinates of $K^{m}$. If now $M>0$ is such that $\varphi(y)$ lies in the real interval $[-M,M]$ for all $y\in X$, then it is clear by construction that $\theta_{d}(\varphi)(x)$ also lies in $[-M,M]$ for all $x\in K^{m}$. ∎

It follows from Lemma 2.3.1 and Proposition 2.2.3 that if $\varphi$ is a bounded function in ${\mathcal{C}}(X)$ one can set

[TABLE]

that is, for $x\in K^{m}$, $\Theta_{d}(\varphi)(x)$ is the mean value at infinity of the function $n\mapsto\theta_{d}(\varphi)(x,n)$. By Proposition 2.2.3, the function $\Theta_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m})$. For $x$ in $K^{m}$, we call $\Theta_{d}(\varphi)(x)$ the local density of $\varphi$ at $x$. More generally, if $\varphi$ is bounded on a neighborhood of some $x\in K^{m}$, then $\Theta_{d}(\varphi)(x)$ can be defined by first extending $\varphi$ by zero outside of this neighborhood and calculate its local density by the above definitions which is clearly independent of the choice of the neighborhood. One should also note that $\Theta_{d}(\varphi)(x)$ is zero when $x$ does not belong to the closure of $X$.

2.3.2 Definition**.**

Let $X$ be a definable subset of $K^{m}$ of dimension $d$ and let $x$ be a point in $K^{m}$. We call the rational number

[TABLE]

the local density of $X$ at $x$, where ${\mathbf{1}}_{X}$ is the characteristic function of $X$ which clearly lies in ${\mathcal{C}}(K^{m})$.

Note that Definition 2.3.2 resembles the definition of the complex and real density as given in the introduction, where instead of the limit $\lim_{r\to 0}$ one takes ${\rm MV}_{\infty}$.

2.3.3 Lemma**.**

Renormalizing with balls instead of with spheres yields the same local density functions $\Theta_{d}$. Precisely, for ${\mathcal{L}}$-definable $X$ of dimension $d$ and for $\varphi$ a bounded function in ${\mathcal{C}}(X)$ one has for $x\in K^{m}$

[TABLE]

where

[TABLE]

[TABLE]

and where $B_{d}(n)$ is the $d$-dimensional ball of radius $q^{-n}$, namely $\{w\in K^{d}\mid|w|\leq q^{-n}\}$, and $B(x,n)$ is the ball $\{y\in K^{m}\mid|x-y|\leq q^{-n}\}$ around $x$ as defined in 1.1. In particular, $\theta^{\prime}_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$ and is $K^{m}$-bounded and thus its ${\rm MV}_{\infty}$ is well-defined.

Proof.

That $\theta^{\prime}_{d}(\varphi)$ lies in ${\mathcal{C}}(K^{m}\times{\mathbf{N}})$ and is $K^{m}$-bounded is proven as Lemma 2.3.1. We have to prove that $\Theta_{d}(\varphi)={\rm MV}_{\infty}(\theta^{\prime}_{d}(\varphi))$, that is, for $x\in K^{m}$, $\Theta_{d}(\varphi)(x)$ is the mean value at infinity of the function $n\mapsto\theta^{\prime}_{d}(\varphi)(x,n)$. It is clear that

[TABLE]

and that

[TABLE]

Now we are done by the following fact, which holds for any real constant $b\not=1$. If a function $f:{\mathbf{N}}\to{\mathbf{R}}$ has a mean value at infinity, then so does $g:{\mathbf{N}}\to{\mathbf{R}}:n\mapsto\frac{1}{1-b}(f(n)-bf(n+1))$, and their mean values at infinity are equal. ∎

2.3.4 Example**.**

Let us note that in the example of 2.1 of points of even valuation in $K$, one gets $\Theta_{1}(X)(0)=\frac{1}{2}$. More generally, if $\Lambda$ is a definable open subgroup of finite index $r$ in $K^{\times}$ and $y$ is a point in $K^{\times}$, we have $\Theta_{1}(\Lambda y)(0)=\frac{1}{r}$. Indeed, it is easily checked that $\Theta_{1}(\Lambda y)(0)$ does not depend on $y$, hence if $y_{1}$, …, $y_{r}$ is a set of representative of $K^{\times}/\Lambda$, we have $1=\Theta_{1}(\cup_{1\leq i\leq r}\Lambda y_{i})(0)=r\Theta_{1}(\Lambda)(0)$.

2.3.5 Proposition**.**

Let $\varphi$ be a bounded function in ${\mathcal{C}}(X)$ and assume $X$ is of dimension $d$. Denote by $\tilde{\varphi}$ the extension of $\varphi$ by zero on $K^{m}$. Then the support of $\tilde{\varphi}-\Theta_{d}(\varphi)$ is contained in an ${\mathcal{L}}$-definable set of dimension $<d$.

Proof.

Suppose $X\subset K^{m}$. Since for $x$ not in $\overline{X}$, for all sufficiently large $n$ one has that $\theta_{d}(\varphi)(x,n)=0$, the support of $\Theta_{d}(\varphi)$ is contained in the closure $\overline{X}$ of $X$ in $K^{n}$. After removing a subset of dimension $<d$ we may assume $X$ is a smooth subvariety and $\varphi$ is locally constant (for example after an iterated application of Lemma 1.4.3), in which case the result is clear. ∎

2.4

For further use we shall give some basic properties of local densities.

2.4.1 Proposition**.**

Let $X$ be definable subset of dimension $d$ of $K^{m}$. Then

[TABLE]

where $\overline{X}$ denotes the closure of $X$.

Proof.

Indeed, by additivity it is enough to prove that $\Theta_{d}(\overline{X}\setminus X)(x)=0$, which follows from the fact that $\Theta_{d}(Y)(x)=0$ when $Y$ is definable of dimension $<d$. ∎

2.4.2 Proposition**.**

*Let $X$ be an ${\mathcal{L}}$-definable set of dimension $d$ and let $M>0$ be a constant. Consider a series of functions $\varphi_{n}:X\to{\mathbf{R}}$, $n\in{\mathbf{N}}$, such that the function $(x,n)\mapsto\varphi_{n}(x)$ lies in ${\mathcal{C}}(X\times{\mathbf{N}})$ and such that $0\leq\varphi_{n}\leq\varphi_{n+1}\leq\dots\leq M$ for all $n$. Then the function $\varphi$ defined as $\sup\varphi_{n}$ lies in ${\mathcal{C}}(X)$ and is bounded. Moreover, *

[TABLE]

and

[TABLE]

for each $n$ and $x$.

Proof.

Clearly the function $\varphi$ is bounded and lies in ${\mathcal{C}}(X)$ by Lemma 2.2.3. Note that

[TABLE]

for each $m$ and $x$ by the Monotone Convergence Theorem. Hence, by the definition of $\theta_{d}$, also

[TABLE]

for each $m$ and $x$. Clearly

[TABLE]

for all $n$, on the whole of $X$, and hence

[TABLE]

by the definition of $\Theta_{d}$. Now the equality $\Theta_{d}(\varphi)(x)=\lim_{n}\Theta_{d}(\varphi_{n})(x)$ follows from (2.4.1), (2.4.2), and the definitions of ${\rm MV}_{\infty}$ and $\Theta_{d}$, by changing the order of limits over $n$ and over $m$. ∎

3 Tangent cones

3.1 Cones

We shall consider the set ${\mathcal{D}}$ of open subgroups of $K^{\times}$ which are of finite index in $K^{\times}$. We order ${\mathcal{D}}$ by inclusion. Note that for each $n>0$, the group $P_{n}$ of the $n$th powers in $K^{\times}$ lies in ${\mathcal{D}}$, and any $\Lambda$ in ${\mathcal{D}}$ equals, as a set, a finite disjoint union of cosets of some $P_{n}$, see Lemma 3.1.1, and is thus ${\mathcal{L}}$-definable. We shall say a certain property (P) holds for $\Lambda$ small enough, if there exists $\Lambda_{0}$ in ${\mathcal{D}}$ such that (P) holds for every $\Lambda\in{\mathcal{D}}$ contained in $\Lambda_{0}$.

Let $\Lambda$ be a subgroup of $K^{\times}$ in ${\mathcal{D}}$. It acts naturally on $K^{n}$ by multiplicative translation $\lambda\cdot z:=\lambda z$, that is, by scalar multiplication on the vector space $K^{n}$. By a $\Lambda$-cone in $K^{n}$ we mean a subset $C$ of $K^{n}$ which is stable under the $\Lambda$-action, that is, $\Lambda\cdot C\subset C$ (note that this implies that $\Lambda\cdot C=C$). More generally, if $x\in K^{n}$, by a $\Lambda$-cone with origin $x$ we mean a subset $C$ of $K^{n}$ such that $C-x$ is stable under the $\Lambda$-action, where $C-x=\{t\in K^{n}\mid t+x\in C\}$. By a local $\Lambda$-cone with origin $x$, we mean a set of the form $C\cap B(x,n)$, with $C$ a $\Lambda$-cone with origin $x$ and $n$ in ${\mathbf{N}}$.

In Lemma 3.1.1 we describe all possible $\Lambda$-cones which are subsets of $K$, which turns out to be very similar to the real situation. In section 3.2 we will show that definable sets in dimension $1$ locally look like local $\Lambda$-cones (Lemma 3.2.1), and similarly in families of definable subsets of $K$ (Corollary 3.2.2). From 3.3 on we will define and study tangent cones and related objects, and formulate one of our main results on the relation between local densities of definable sets and of their tangent cones, viewed with multiplicities (the $p$-adic analogue of Thie’s result), see Theorem 3.6.2.

3.1.1 Lemma**.**

Let $C\subset K$ be a set. Then $C$ is a $\Lambda$-cone for some $\Lambda$ in ${\mathcal{D}}$ if and only if it is either the empty set or it is a finite disjoint union of sets of the form $\lambda P_{n}$ with $n>0$ and $\lambda\in K$. Hence, any cone $C\subset K$ is a definable set.

Proof.

Clearly the empty set is a $\Lambda$-cone for all $\Lambda$ and $\lambda\cdot P_{n}$ is a $\Lambda$-cone for $\Lambda\subset P_{n}$, and similarly for their finite unions. Now let $C$ be a nonempty $\Lambda$-cone for some $\Lambda$ in ${\mathcal{D}}$. Either $C=\{0\}$ and we are done, or, up to replacing $C$ by $tC$ for some nonzero $t\in K$, we may suppose that $1\in C$. But then $\Lambda\subset C$ and $C\setminus\Lambda$ is still a $\Lambda$-cone. Since the index of $\Lambda$ in $K^{\times}$ is finite, it follows by a finite process that $C$ consists of a finite union of sets of the form $\mu\Lambda$ with $\mu\in K$. It remains to prove that $\Lambda$ itself is a finite disjoint union of sets of the form $\lambda P_{n}$, for some $n\in{\mathbf{N}}$ and some $\lambda\in K^{\times}$. Since $\Lambda$ has finite index in $K^{\times}$, it must contain an open neighborhood $U=1+{\mathcal{M}}_{K}^{\ell}$ of $1$ for some $\ell>0$ and with ${\mathcal{M}}_{K}$ the maximal ideal of $R$. Let $\pi_{K}$ be a uniformizer of $R$. Since $\Lambda$ has finite index in $K^{\times}$, there exists $n_{1}>0$ such that $\pi_{K}^{n_{1}}$ lies in $\Lambda$. Now let $n$ be a big enough multiple of $n_{1}$ such that $t^{n}\in U$ for all $t\in R^{\times}$. Then clearly $P_{n}\subset\Lambda$ and we are done since $P_{n}$ has finite index in $K^{\times}$ and hence also in $\Lambda$. ∎

3.2 Local conic structure of definable sets

Let $X$ be a definable subset of $K^{n}$ and let $x$ be a point in $K^{n}$. We denote by $\pi_{x}:K^{n}\setminus\{x\}\rightarrow{\mathbf{P}}^{n-1}(K)$ the function which to a point $z\not=x$ assigns the line containing $x$ and $z$. That is, for $x=0$, $\pi_{0}:K^{n}\setminus\{0\}\rightarrow{\mathbf{P}}^{n-1}(K)$ is the natural projection, and, for nonzero $x$, the map $\pi_{x}$ is the composition of $\pi_{0}$ with the translation $K^{n}\setminus\{x\}\to K^{n}\setminus\{0\}:y\mapsto y-x$. Furthermore we denote by

[TABLE]

the restriction of $\pi_{x}$ to $X\setminus\{x\}$.

3.2.1 Lemma**.**

Let $Y$ be a definable subset of $K$. Then there exist $\Lambda$ in ${\mathcal{D}}$ and a definable function $\gamma:K\to{\mathbf{N}}$ such that $Y\cap B(y,\gamma(y))$ is a local $\Lambda$-cone with origin $y$, for all $y\in K$. If one writes $Y$ as a finite disjoint union of cells $Y_{i}$ with cosets $\lambda_{i}P_{n_{i}}$, then one can take $\Lambda=P_{N}$ with $N=\operatorname*{lcm}(n_{i})_{i}$.

Proof.

The definability of $\gamma$ is not an issue by the definability of the conditions of being a local $\Lambda$-cone with origin $y$ and so on. By definition, a finite union of local $\Lambda$-cones is again a local $\Lambda$-cone. Hence, up to a finite partition using cell decomposition, we may suppose that $Y$ is a cell. Thus, $Y$ is of the form

[TABLE]

for some constants $n>0$, $\lambda,c$ in $K$, $\alpha,\beta$ in $K^{\times}$, and $\square_{i}$ either $<$ or no condition. Up to a transformation $t\mapsto t-c$ we may suppose that $c=0$. We may exclude the trivial case that $Y$ is a singleton, that is, we may suppose that $\lambda\not=0$. Then $Y$ is open, and moreover, $Y$ is closed if and only if ${\operatorname{\square}\nolimits}_{2}$ is $<$. In the case that ${\operatorname{\square}\nolimits}_{2}$ is no condition, then the closure of $Y$ equals $Y\cup\{0\}$. Take $y\in K$. If $y$ lies outside the closure of $Y$, then $Y\cap B(y,n)$ is empty for sufficiently large $n$, and the empty set is a $\Lambda$-cone for any $\Lambda$ in ${\mathcal{D}}$. Also, if $y$ lies in the interior of $Y$, then $Y\cap B(y,n)$ is a ball around $y$ for sufficiently large $n$, and hence it is a local $\Lambda$-cone with origin $y$, for any $\Lambda$ in ${\mathcal{D}}$. Finally, if $y=0$ and $y$ lies in the closure of $Y$, then $Y\cap B(y,n)=\lambda P_{n}\cap B(y,n)$ for sufficiently large $n$, which is clearly a local $\Lambda$-cone with origin $y$ for any $\Lambda$ contained in $P_{n}$. ∎

The following two corollaries of Lemma 3.2.1 are immediate.

3.2.2 Corollary**.**

Let $Y$ be a definable subset of $K^{m+1}$. For each $x\in K^{m}$ write $Y_{x}$ for $\{t\in K\mid(x,t)\in Y\}$. Then there exist $\Lambda$ in ${\mathcal{D}}$ and a definable function $\gamma:K^{m+1}\to{\mathbf{N}}$ such that $Y_{x}\cap B(t,\gamma(x,t))$ is a local $\Lambda$-cone with origin $t$, for all $(x,t)\in K^{m+1}$. If one writes $Y$ as a finite disjoint union of cells $Y_{i}$ with cosets $\lambda_{i}P_{n_{i}}$, then one can take $\Lambda=P_{N}$ with $N=\operatorname*{lcm}(n_{i})_{i}$.

We will most often use the following variant of Corollary 3.2.2, which can be proved by working on affine charts.

3.2.3 Corollary**.**

Let $X$ be a definable subset of $K^{n}$ and let $x$ be a point in $K^{n}$. Then there exist a definable function $\alpha_{x}:{\mathbf{P}}^{n-1}(K)\rightarrow{\mathbf{N}}$, that is, $\alpha_{x}$ is definable on each affine chart of ${\mathbf{P}}^{n-1}(K)$, and a group $\Lambda$ in ${\mathcal{D}}$ such that

[TABLE]

is a local $\Lambda$-cone with origin $x$ for every $\ell$ in ${\mathbf{P}}^{n-1}(K)$. Moreover, $\Lambda$ can be taken independently of $x$, and one can ensure that $(x,\ell)\mapsto\alpha_{x}(\ell)$ is a definable function from $K^{n}\times{\mathbf{P}}^{n-1}(K)$ to ${\mathbf{N}}$.

We shall call a subgroup $\Lambda$ in ${\mathcal{D}}$ satisfying the first condition in Corollary 3.2.3 adapted to $(X,x)$, and if moreover $\Lambda$ is adapted to $(X,x)$ for all $x\in K^{n}$, then we call $\Lambda$ adapted to $X$.

3.3 Tangent cones

Now, if $X$ is a definable subset of $K^{n}$, $x$ a point of $K^{n}$, and $\Lambda$ in ${\mathcal{D}}$, we define the tangent $\Lambda$-cone to $X$ at $x$ as

[TABLE]

By construction $C_{x}^{\Lambda}(X)$ is a closed, definable, $\Lambda$-cone, and, for any $n\in{\mathbf{N}}$, $C_{x}^{\Lambda}(X)=C_{x}^{\Lambda}(X\cap B(x,n))$. Furthermore, for definable $X,Y\subset K^{n}$ and for $\Lambda^{\prime}\subset\Lambda$ in ${\mathcal{D}}$, one has

[TABLE]

[TABLE]

[TABLE]

Although the previous inclusion might be strict, $\dim(C_{x}^{\Lambda}(X))$ does not depend on $\Lambda\in\mathcal{D}$ by Lemma 3.5.1. We comment some more on the previous inclusion in the following remarks.

3.3.1 Remark*.*

Let $X$ be a local $\Lambda$-cone with origin $x$ in $K^{n}$. Thus, there exist $n$ in ${\mathbf{N}}$ and $C$ a $\Lambda$-cone with origin $x$ such that $X=C\cap B(x,n)$. In this case for any $\Lambda^{\prime}\subset\Lambda\in{\mathcal{D}}$, one has

[TABLE]

Indeed, since $X=C\cap B(0,n)$, we have $C_{x}^{\Lambda^{\prime}}(X)=C_{x}^{\Lambda^{\prime}}(C)$. But we also have $C_{x}^{\Lambda^{\prime}}(C)=C_{x}^{\Lambda}(C)$ ($=C$, when $C$ is closed).

We indicate why $C_{x}^{\Lambda}(C)\subset C_{x}^{\Lambda^{\prime}}(C)$. Assuming $x=0$ for simplicity, let $u\in C_{0}^{\Lambda}(C)$ and $i\in{\mathbf{N}}$, $z\in C$, $\lambda\in\Lambda$, with ${\rm ord}(z)>i$ and ${\rm ord}(\lambda z-u)>i$. We have $z\in C$ and thus $\lambda z\in C$. Now let $\lambda^{\prime}\in\Lambda^{\prime}$ small enough to ensure that ${\rm ord}(\lambda^{\prime}\lambda z)>i$. Then denoting $z^{\prime}=\lambda^{\prime}\lambda z$, one has $z^{\prime}\in C$. From ${\rm ord}(z^{\prime})>i$ and ${\rm ord}((1/\lambda^{\prime})z^{\prime}-u)>i$, we see that $u\in C_{0}^{\Lambda^{\prime}}(C)$.

Finally we indicate why $C_{x}^{\Lambda}(C)=C$, when $C$ is closed. As $C$ is stable by the $\Lambda$-action, the inclusion $C\subset C_{x}^{\Lambda}(C)$ is obvious. On the other hand, assuming again $x=0$, if $u\in C_{x}^{\Lambda}(C)$, for all $i\in{\mathbf{N}}$, there exist $z\in C$ and $\lambda\in\Lambda$ such that ${\rm ord}(z)>i$ and ${\rm ord}(\lambda z-u)>i$. We can then construct a sequence of points $\lambda z\in C$ with limit $u$, this shows that $u\in C$, since $C$ is closed.

3.3.2 Remark*.*

When $X$ is a definable subset of $K$ and $x$ a point of $K$, by Lemma 3.2.1, $X$ is a local $\Lambda$-cone at $x$ with origin $x$, for some $\Lambda\in{\mathcal{D}}$. By the above remark, for every $\Lambda^{\prime}\subset\Lambda$, and still in the one dimensional case that $X\subset K$, one has $C_{x}^{\Lambda^{\prime}}(X)=C_{x}^{\Lambda}(X)$.

We cannot expect in general that for $X$ a definable subset of $K^{n}$, $n>1$, $X$ is a local $\Lambda$-cone for some $\Lambda\in{\mathcal{D}}$, but one may at least ask, as it is the case for $n=1$, whether the stability property: “there exists $\Lambda\in{\mathcal{D}}$ such that for any $\Lambda^{\prime}\in{\mathcal{D}}$, $\Lambda^{\prime}\subset\Lambda$, one has $C_{x}^{\Lambda^{\prime}}(X)=C_{x}^{\Lambda}(X)$” still holds for $n>1$. The answer to that question is yes, as we shall show in Theorem 5.6.1.

3.4 More on $\varepsilon$-analytic functions

The following is the $p$-adic analogue of Proposition 1.7 of [29].

3.4.1 Proposition**.**

Let $f:U\to K^{n-d}$ be a definable $\varepsilon$-analytic function on a nonempty open subset $U$ of $K^{d}$, $0\leq d\leq n$. Let $\Gamma$ be the graph of $f$ and let $z$ be in $\overline{\Gamma}$. Then, for any group $\Lambda$ in ${\mathcal{D}}$

[TABLE]

Proof.

We may suppose that $z=0$. Choose $\Lambda$ in ${\mathcal{D}}$. Since

[TABLE]

by definition of $C_{0}^{\Lambda}(\cdot)$, we may assume that $\Lambda=K^{\times}$. We may also suppose that $\varepsilon=1$, after rescaling. Suppose by contradiction that there is $(x_{0},y_{0})$ in $C_{0}^{K^{\times}}(\Gamma)$ with $|y_{0}|>|x_{0}|+\delta$ for some $\delta>0$. Let $\Gamma^{\prime}$ be the intersection of $\Gamma$ with the open subset $\{(x,y)\in K^{d+(n-d)}\mid|y|>|x|+\delta\}$. By our assumption on $(x_{0},y_{0})$ and by the definition of $C_{0}^{K^{\times}}(\Gamma)$, the set $\Gamma^{\prime}$ is nonempty and [math] lies in $\overline{\Gamma}^{\prime}\setminus\Gamma^{\prime}$. Apply the Curve Selection Lemma 1.4.4 to the set $\Gamma^{\prime}$ and the point [math]. This way we find power series $g_{i}$ over $K$ in one variable for $i=1,\ldots,n$, converging on $R$, such that $g(0)=0$ and $g(R\setminus\{0\})\subset\Gamma^{\prime}\setminus\{0\}$. But this is in contradiction with Lemma 1.4.5. Indeed, the tangent line $\ell^{\prime}_{r}$ at $r\not=0$ is of the form $g(r)+K\cdot t_{r}$ with some $t_{r}\in K^{n}$ satisfying $|y(t_{r})|\leq|x(t_{r})|$ by $\varepsilon$-analyticity of $f$ and the chain rule for differentiation, where $x(t_{r})=(t_{r1},\ldots,t_{rd})$ and $y(t_{r})=(t_{rd+1},\ldots,t_{rn})$. Hence, the limit $\ell^{\prime}_{0}$ of the $\ell^{\prime}_{r}$ for $0\not=r\ \to 0$ is of the same form $g(0)+K\cdot t_{0}$ for some $t_{0}\in K^{n}$ with $|y(t_{0})|\leq|x(t_{0})|$. On the other hand, the line $\ell_{r}$ for $r\not=0$ connecting $g(0)$ with $g(r)$ is of the form $g(r)+K\cdot u_{r}$ with some $u_{r}\in K^{n}$ satisfying $|y(u_{r})|>|x(u_{r})|+\delta$. Hence, the limit line $\ell_{0}$ of the $\ell_{r}$ for $r\to 0$ has the same description, which contradicts Lemma 1.4.5 and the description of $\ell^{\prime}_{0}$.

∎

3.4.2 Corollary**.**

With the data and the notation of Proposition 3.4.1, let $x$ be in $\overline{U}$. Then there are only finitely many points in $\overline{\Gamma}$ which project to $x$ under the coordinate projection $K^{d}\times K^{n-d}\to K^{d}$.

Proof.

Suppose by contradiction that there are infinitely many such points. Then the dimension of $\overline{\Gamma}\cap(\{x\}\times K^{n-d})$ is $>0$. Thus, there exists $z\in\overline{\Gamma}$ such that $C_{z}^{\Lambda}(\overline{\Gamma}\cap(\{x\}\times K^{n-d}))$ is of dimension $>0$, which is in contradiction to Proposition 3.4.1. ∎

3.5 Deformation to the tangent cone

Let $X$ be a definable subset of $K^{n}$ and let $x$ be a point of $K^{n}$. Fix a subgroup $\Lambda$ in ${\mathcal{D}}$. We consider the definable set ${\mathcal{D}}(X,x,\Lambda)$ in $K^{n}\times\Lambda$ defined as

[TABLE]

and its closure

[TABLE]

in $K^{n}\times K$. In $\overline{{\mathcal{D}}(X,x,\Lambda)}$ one finds back the cone $C_{x}^{\Lambda}(X)$. Indeed, one has $\overline{{\mathcal{D}}(X,x,\Lambda)}\cap(K^{n}\times\{0\})=C_{x}^{\Lambda}(X)\times\{0\}$, which we identify with $C_{x}^{\Lambda}(X)$.

3.5.1 Lemma**.**

If $X$ is of dimension $d$, then ${\mathcal{D}}(X,x,\Lambda)$ is of dimension $d+1$ and $C_{x}^{\Lambda}(X)$ is of dimension $\leq d$. Moreover, $\dim(C_{x}^{\Lambda}(X))$ does not depend on the choice of $\Lambda\in\mathcal{D}$.

Proof.

We may suppose that $X$ is nonempty. Consider the projection

[TABLE]

Since $p$ is surjective and has fibers of dimension $1$, we get that ${\mathcal{D}}(X,x,\Lambda)$ is of dimension $d+1$. The cone $C_{x}^{\Lambda}(X)$ is contained in $\{0\}\cup(\overline{{\mathcal{D}}(X,x,\Lambda)}\setminus{\mathcal{D}}(X,x,\Lambda))$. Hence, $C_{x}^{\Lambda}(X)$ is of dimension $\leq d$. The last statement follows from

[TABLE]

whenever $\Lambda^{\prime}\subset\Lambda$ is in ${\mathcal{D}}$ and when one writes $\Lambda$ as a finite union of cosets $\bigcup_{i}\mu_{i}\Lambda^{\prime}$ of $\Lambda^{\prime}$ in $\Lambda$. ∎

3.6 Multiplicities on the tangent cones

Let $X$ be a definable subset of $K^{n}$ of dimension $d$, let $x$