1 Introduction

In this paper, we continue our study (started in [4, 10, 12,13,14], compare also [5]) of certain very special settings of the classical Whitney’s smooth extension problem (see [1,2,3, 6,7,8]). Let’s recall shortly the general setting of this problem, and its current status.

Let \(E\subset B^n \subset {{\mathbb {R}}}^n\) be a closed subset of the unit ball \(B^n\), and let \({\bar{f}}\) be a real function, defined on E. Can \({\bar{f}}\) be extended to a \(C^d\)-smooth f on \({\mathbb R}^n\), and, if the extension is possible, what is the minimal \(C^d\)-norm of f?

Recent exciting developments in the general Whitney problem (see [1,2,3] and references therein), provide essentially a complete answer to the general Whitney extension problem in any dimension. In particular, the results of [1,2,3]), provide an important information on this problem, which was earlier available only in dimension one. The “finiteness principle” achieved in this recent work, claims that, as in classical Whitney’s results in dimension one [7], it is enough to check only finite subsets of Z with cardinality bounded in terms of n and d only. There is also an algorithmic way to estimate the minimal extension norm for any finite Z.

However, a possibility of an explicit answer, as in dimension one, through a kind of multi-dimensional divided finite differences, remains an open problem.

Let us now describe the setting of [4, 12,13,14], which we use below. Let \(Z\subset B^n \subset {{\mathbb {R}}}^n\) be a closed subset of the unit ball \(B^n\). In [4, 12,13,14], we look for \(C^{d+1}\)-smooth functions \(f:B^n\rightarrow {{\mathbb {R}}}\), vanishing on Z. Such \(C^{d+1}\)-smooth (and even \(C^\infty \)) functions f always exist, since any closed set Z is the set of zeroes of a \(C^\infty \)-smooth function.

We normalize the extensions f requiring \(\mathrm{max\,}_{B^n}|f|=1\), and ask for the minimal possible norm of the last derivative \(||f^{(d+1)}||\), which we call the d-rigidity \(\mathcal{R}\mathcal{G}_d(Z)\) of Z. In other words, for each normalized \(C^{d+1}\)-smooth function \(f:B^n\rightarrow {{\mathbb {R}}},\) vanishing on Z, we have

$$\begin{aligned} ||f^{(d+1)}||\ge \mathcal{R}\mathcal{G}_d(Z), \end{aligned}$$

and \(\mathcal{R}\mathcal{G}_d(Z)\) is the maximal number with this property. Here the norm \(||f^{(d+1)}||\) is the maximum over \(x\in B^n\) of the point-wise norms \(||f^{(d+1)}(x)||\), the last being the sum of the absolute values of all the partial derivatives of order \(d+1\) of f at x.

Our previous papers [4, 12,13,14], related to smooth rigidity, provide certain bounds on \(\mathcal{R}\mathcal{G}_d(Z)\), in terms of the fractal geometry of Z. We use some of these results below. In order to compare the results of [4, 12,13,14] with the general results, available today in Whitney’s extension theory, let’s make the following remark:

Of course, the results of [2, 3] provide, in principle, an algorithmic way to estimate also our quantities \(\mathcal{R}\mathcal{G}_d(Z)\), for any closed \(Z\subset B^n\) (via considering finite subsets of Z of bounded cardinality). However, our goal in the present paper, as well as in our previous papers [4, 10,11,12,13,14], related to smooth rigidity, is somewhat different: we look for an explicit answer in terms of simple, and directly computable geometric (or topological) characteristics of Z.

Now we finally come to the setting of the problem in the present paper, and to the new results below. In the present paper, we consider, as above, \(C^{d+1}\)-smooth functions \(f:B^n\rightarrow {\mathbb R}\). But, in contrast to [4, 12,13,14], we do not consider zero sets of f. Instead, we assume some geometric conditions on the critical points and values of f, and, as above, derive in conclusion some lower bounds on \(||f^{(d+1)}||\).

More accurately, put, as usual,

$$\begin{aligned} \nabla f(x)=\left( \frac{\partial f}{\partial x_1}(x),\ldots ,\frac{\partial f}{\partial x_n}(x)\right) \end{aligned}$$

at each point \(x\in B^n\).

The point \(x\in B^n\) is called a critical (or a singular) point for f, if the vector equation \(\nabla f(x)=0\) is satisfied. The real \(\nu \) is called a critical (or a singular) value of f if \(\nu =f(x)\) for a certain critical point x of f.

Remark

Prescribing the set of the critical points of f is not that immediate as for the zeroes set of f (which can be any closed subset of \(B^n\)). Indeed, consider, for instance \(Z=S=\{x_1^2+x_2^2=\frac{1}{4}\}\) being the circle of radius \(\frac{1}{2}\). Then for any smooth f on \(B^2\) with \(\nabla f\) vanishing on S the restriction of f to S is a constant. Consequently, there is a critical point of f inside S.

For \(Z=T^2 \subset B^3\) being the standard torus, any smooth f on \(B^2\) with \(\nabla f\) vanishing on Z must have some critical points in the interior of the torus, by similar topological reasons.

Let us now state the main results of this paper. From now on we always assume that the dimension n, as well as an integer \(d\ge 1\) are fixed. We consider two settings of the problem:

First, in Sect. 2, we look only on the geometry of the critical points of f, not taking into account the critical values, and not requiring any non-degeneracy. Here we can apply, with almost no changes, the results of [4, 12,13,14], and obtain a series of new results, where the set Z of zeroes of f is replaced by the set \(\Sigma \) of the critical points of f.

Next, in Sect. 3, we look only on the geometry of the critical values of f, completely ignoring the geometry of the critical points. Here, the results are obtained by a kind of “backward reading” of the old results of [9, 15]. Still, these new results seem to provide an important new information on the rigidity properties of smooth functions.

The authors would like to thank the referee for valuable remarks and suggestions, which allowed us to significantly clarify and improve the presentation.

2 Geometry of critical points

Let \(\Sigma \subset B^n \subset {{\mathbb {R}}}^n\) be a closed subset of the unit ball \(B^n\). We look for \(C^{d+1}\)-smooth functions \(f:B^n\rightarrow {{\mathbb {R}}}\), with the gradient f vanishing on \(\Sigma \).

We normalize these functions f requiring \(\mathrm{max\,}_{B^n}|f|=1\), on one side, and \(f(x_0)=0\) for certain \(x_0\in B^n\), on the other side, and call the set of such functions \(U(d, \Sigma )\). Then we ask for the minimal possible norm of the last derivative \(||f^{(d+1)}||\), which we call the \(d^1\)-rigidity \(\mathcal{R}\mathcal{G}^1_d(\Sigma )\) of \(\Sigma \).

To avoid possible misreading, let’s provide a formal definition:

Definition 2.1

The \(d^1\)-rigidity \(\mathcal{R}\mathcal{G}^1_d(\Sigma )\) of \(\Sigma \) is defined as

$$\begin{aligned} \mathcal{R}\mathcal{G}^1_{d}(\Sigma ) = \mathrm{max\,}R, \ \ with \ \ ||f^{(d+1)}|| \ge R, \ \ {for \ each} \ \ f \in U(d, \Sigma ). \end{aligned}$$

In other words, for each normalized \(C^{d+1}\)-smooth function \(f:B^n\rightarrow {{\mathbb {R}}},\) with \(\nabla f\) vanishing on \(\Sigma \), we have

$$\begin{aligned} ||f^{(d+1)}||\ge \mathcal{R}\mathcal{G}^1_d(\Sigma ), \end{aligned}$$

and \(\mathcal{R}\mathcal{G}^1_d(\Sigma )\) is maximal with respect to this property.

Now, we show that the “rigidities” \(\mathcal{R}\mathcal{G}^1_d(\Sigma )\) and \(\mathcal{R}\mathcal{G}_d(\Sigma )\) are “subordinated”: the second bounds from below the first (up to a constant):

Theorem 2.1

$$\begin{aligned} \mathcal{R}\mathcal{G}^1_{d}(\Sigma )\ge \frac{1}{\sqrt{2n}} \mathcal{R}\mathcal{G}_d(\Sigma ). \end{aligned}$$

Proof

Pick a function \(f \in U(d, \Sigma )\). Now we notice that if \(\nabla f\) vanishes on \(\Sigma \), then each of the partial derivatives \(\frac{\partial f}{\partial x_i}, i=1,\ldots , n,\) vanishes on \(\Sigma \). On the other hand, for the normalized f, at least one of these partial derivatives, say, \(f'_1=\frac{\partial f}{\partial x_1}\), attains sufficiently big value, say, \(\frac{1}{\sqrt{2n}}\), inside the ball \(B^n\).

Indeed, consider the straight segment \(\ell =[w^0,w^1]\), from one of the points \(w^0=(w^0_1,\ldots ,w^0_n)\), where f vanishes, to one of the points \(w^1=(w^1_1,\ldots ,w^1_n)\), where \(|f(w^1)|=1\). Let \(\phi : [0,1]\rightarrow \ell \) be the affine parametrization of \(\ell \), given for \(t\in [0,1]\) by

$$\begin{aligned} \phi (t)=w^0 + tV, \ \ V=(v_1,\ldots ,v_n):=(w^1_1-w^0_1,\ldots ,w^1_n-w^0_n). \end{aligned}$$

For \(t \in [0,1]\) we put \(g(t)=f(\phi (t))\). Replacing, if necessary, f with \(-f\), we can assume that \(f(w^1)=g(1)=1\), and thus we have

$$\begin{aligned} 1= & {} \int _0^1 \frac{\textrm{d}g}{\textrm{d}t} \textrm{d}t = \int _0^1 \textrm{d}t \sum _{i=1}^n \frac{\partial f}{\partial x_i}(\phi (t)) \frac{\textrm{d}x_i}{\textrm{d}t} \\= & {} \sum _{i=1}^n v_i \int _0^1 \frac{\partial f}{\partial x_i}(\phi (t))\textrm{d}t = V\cdot \Omega \le ||V||||\Omega ||, \end{aligned}$$

where \(\Omega = (\omega _1,\ldots ,\omega _n),\) with \(\omega _i=\int _0^1 \frac{\partial f}{\partial x_i}(\phi (t))\textrm{d}t, \ \ \ i=1,\ldots ,n,\) and \(V\cdot \Omega \) denotes the scalar product of these vectors.

Thus we have \(||V||||\Omega ||\ge 1\), while \(||V||\le 2,\)V being the vector joining two points in \(B^n\). We conclude that \(||\Omega ||\ge 1/2,\) and hence at least one of its coordinates \(\omega _i\) (say, the first one) is not smaller than \(\frac{1}{\sqrt{2n}}\). Finally, \(\omega _1=\int _0^1 \frac{\partial f}{\partial x_1}(\phi (t))\textrm{d}t \ge \frac{1}{\sqrt{2n}}\) implies that for some \(t\in [0,1]\) we have \(\frac{\partial f}{\partial x_1}(\phi (t)) \ge \frac{1}{\sqrt{2n}}\).

Therefore, the function \(f'_1\) vanishes on \(\Sigma \), and its maximum on \(B^n\) is at least \(\frac{1}{\sqrt{2n}}\). According to the definition of \(\mathcal{R}\mathcal{G}_d(\Sigma )\) given above, we conclude that

$$\begin{aligned} ||f^{(d+1)}||\ge ||(f'_1)^{(d)}|| \ge \frac{1}{\sqrt{2n}} \mathcal{R}\mathcal{G}_d(\Sigma ), \end{aligned}$$

which implies

$$\begin{aligned} \mathcal{R}\mathcal{G}^1_{d}(\Sigma )\ge \frac{1}{\sqrt{2n}} \mathcal{R}\mathcal{G}_d(\Sigma ). \end{aligned}$$

This completes the proof of Theorem 2.1. \(\square \)

Our main conclusion is that the results of [4, 12,13,14] are directly applicable to the partial derivatives \(\frac{\partial f}{\partial x}\). These results are given in terms of the metric “density” of \(\Sigma \), of its “Remez Constant”, its metric entropy (the asymptotic behavior of the covering numbers), and in terms of the topology of \(\Sigma \). We address the interested reader to [4, 12,13,14].

3 Critical values

This section presents the main new results of the present paper. Because of the nature of the results and of the proofs, we extend the original setting above from functions f on \(B^n\) to mappings \(f:B^n_r\rightarrow B^m, \ m\le n, \ r>0.\) Here \(B^n_r\) is the ball of radius r centered at the origin of \({{\mathbb {R}}}^n, \ \ B^n:=B^n_1\).

3.1 Definitions and density bounds

We start with recalling some standard definitions, as well as one of the central results from [9, 15], providing an upper bound for the metric entropy of critical and near-critical values of smooth mappings.

For a mapping \(f:B^n_r\rightarrow B^m, \ m\le n, \ r>0\), and for each \(x\in B_r^n\), the differential df(x) is a linear mapping \(df(x): {{\mathbb {R}}}^n\rightarrow {{\mathbb {R}}}^m.\) The image \(df(x)(B^n)\) of the unit ball \(B^n\) in \({{\mathbb {R}}}^n\) is an ellipsoid in \({{\mathbb {R}}}^m\). We denote the semi-axes of this ellipsoid by

$$\begin{aligned} 0 \le \lambda _1(x)\le \lambda _2(x)\le \cdots \le \lambda _m(x), \end{aligned}$$

and put

$$\begin{aligned} \Lambda (f,x)= \ (\lambda _1(x),\lambda _2(x), \cdots , \lambda _m(x)). \end{aligned}$$

Accordingly, we define the set of near-critical points of f, and the set of near-critical values of f as follows:

Definition 3.1

For a given

$$\begin{aligned} \Lambda =(\lambda _1,\lambda _2, \ldots , \lambda _m), \ 0 \le \lambda _1(x)\le \lambda _2(x)\le \cdots \le \lambda _m(x), \end{aligned}$$

the set \(\Sigma (f,\Lambda )\) of \(\Lambda \)-near-critical points of f, and the set \(\Delta (f,\Lambda )\) of \(\Lambda \)-near-critical values of f are defined as follows:

$$\begin{aligned} \Sigma (f,\Lambda ) = \{x\in B_r^n, \ \lambda _i(x) \le \lambda _i, \ i=1,\ldots ,m \}, \ \Delta (f,\Lambda )=f(\Sigma (f,\Lambda )). \end{aligned}$$

Let’s now recall the definition of two of the most important parameters involved in our further calculations:

Definition 3.2

For a natural d the Taylor constant \(R_d(f)\) is defined as

$$\begin{aligned} R_d(f)=\frac{\mathrm{max\,}_{x\in B_r^n} ||f^{(d)}(x)||}{d!}r^d = \frac{||f^{(d)}||}{d!}r^d. \end{aligned}$$

Definition 3.3

For a compact set S in a metric space Y, and for \(\epsilon >0\) the covering number \(M(\epsilon ,S)\) is defined as the minimal number of the \(\epsilon \)-balls in Y, covering S.

Now, we recall one of the main result of [9, 15] which is crucial for our results below. For a convenience reasons we always put \(\lambda _0=1\).

Theorem 3.1

(Theorem 9.2, [15])

Let \(f:B^n_r\rightarrow {{\mathbb {R}}}^m\) be a \(C^d\)-smooth mapping. Then for a given

$$\begin{aligned} \Lambda =(\lambda _1,\lambda _2, \ldots , \lambda _m) \end{aligned}$$

as above, and for \(\epsilon >0\) we have

$$\begin{aligned}{} & {} M(\epsilon , \Delta (f,\Lambda )) \le c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i, \ \ \ \ \ \ \ \epsilon \ge R_d(f). \\{} & {} M(\epsilon , \Delta (f,\Lambda )) \le c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i \left( \frac{R_d(f)}{\epsilon }\right) ^{\frac{n-i}{d}}, \ \ \ \ \ \ \ \epsilon \le R_d(f), \end{aligned}$$

where c is a constant depending only on n, m, d, r.

Many related results and applications of Theorem 3.1 can be found in [9, 15] and references therein.

3.2 A new result: rigidity via critical values

Our goal now is “to read Theorem 3.1 in the opposite direction”, and to bound \(R_d(f)\) from below in terms of the covering numbers \(M(\epsilon , \Delta (f,\Lambda ))\). An accurate setting of the problem is as follows:

We consider \(C^d\)-smooth mappings \(f:B^n_r\rightarrow {{\mathbb {R}}}^m\), with n, m, d, r fixed. We also fix \(\Lambda =(\lambda _1,\lambda _2, \ldots , \lambda _m)\) as above.

Next we fix a compact set \(D \subset B^m_1 \subset {{\mathbb {R}}}^m\). Our goal is to produce, for any d-smooth f as above, with \(D \subseteq \Delta (f,\Lambda )\), an explicit lower bound for the Taylor constant \(R_d(f)\). We want the answer to be in terms of \(n,m,d,r,\Lambda ,\) and of the metric entropy of D only.

The required bound is provided by Theorem 3.2 (see also Proposition 3.1) below. To state and prove this theorem, we need some preliminary definitions and results.

Definition 3.4

For given D and \(\Lambda \) as above the subset \(E(D,\Lambda ) \subset {{\mathbb {R}}}_+\) consists of all \(\epsilon >0\) for which

$$\begin{aligned} M(\epsilon , D) > c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i. \end{aligned}$$
(3.1)

The set \(E(D,\Lambda )\) plays an important role in our considerations. Only for \(\epsilon \in E(D,\Lambda )\) the \(\epsilon \) - covering number of D is big enough to provide a non-trivial lower bound on \(R_d(f)\). Indeed, the first inequality of Theorem 3.1 describes, essentially, the case of f being a polynomial of degree \(d-1\). Therefore, for D contained in the set of \(\Lambda \) - near critical values of such f the set \(E(D,\Lambda )\) is empty.

The geometry of \(E(D,\Lambda )\) may be quite complicated, especially for “fractal” sets D, containing “lacunary” geometric scales. We plan to address these questions separately, while in the present paper we only give a simple geometric bound on \(E(D,\Lambda )\), in terms of the covering numbers \(M(\epsilon ,B^m_1)\) of the unite ball \(B^m_1\) in \({{\mathbb {R}}}^m\). Let \({\bar{\epsilon }} = {\bar{\epsilon }}_{n,m,d,r}\) be the biggest \(\epsilon \) for which \(M(\epsilon ,B^m_1)\ge [c]+1\), where \(c=c(n,m,d,r)\) is the constant of Theorem 3.1.

Lemma 3.1

For each compact set \(D \subset B^m_1 \subset {{\mathbb {R}}}^m\) and for each \(\Lambda \) we have

$$\begin{aligned} E(D,\Lambda ) \subset [0,{\bar{\epsilon }}]. \end{aligned}$$

Proof

The right-hand side of (3.1) is a decreasing function in \(\epsilon \), which is not smaller that the first term \(c=c(n,m,d,r)\) in the sum (recall that \(\lambda _0:=1)\). By the definition of \({\bar{\epsilon }}\) we get for each \(\epsilon > {\bar{\epsilon }}\)

$$\begin{aligned} M(\epsilon , D) \le M(\epsilon ,B^m_1) \le c \le c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i. \end{aligned}$$

Consequently, no \(\epsilon > {\bar{\epsilon }}\) can belong to \(E(D,\Lambda )\). \(\square \)

The following observation provides an initial form of the required lower bound for \(R_d(f)\):

Proposition 3.1

If \(D \subseteq \Delta (f,\Lambda )\) for some f as above, then for each \(\epsilon \in E(D,\Lambda )\) we have \(\epsilon <R_d(f)\). In particular, we have \(R_d(f) \ge \sup _{\epsilon \in E(D,\Lambda )}\epsilon \).

Proof

If for a certain \(\epsilon >0\) and for f as above we have \(\epsilon \ge R_d(f)\), then by the first inequality of Theorem 3.1 we would have

$$\begin{aligned} M(\epsilon ,D) \le M(\epsilon , \Delta (f,\Lambda )) \le c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i, \end{aligned}$$

so by Definition 3.4 this \(\epsilon \) does not belong to \(E(D,\Lambda )\). \(\square \)

If \(E(D,\Lambda ) \ne \emptyset \), then the lower bound of Proposition 3.1 provides a non-trivial information, but in many cases it is far from being sharp. This concerns especially the situations where the covering numbers of D grow fast with \(\epsilon \rightarrow 0\), while the bound of Proposition 3.1 is at most \({\bar{\epsilon }}\), by Lemma 3.1. Compare Theorem 3.2, and the specific examples below.

Now let us fix \(\epsilon \in E(D,\Lambda )\). For an auxiliary variable \(\eta \ge \epsilon \) consider the following equation in \(\eta \):

$$\begin{aligned} M(\epsilon , D)) = c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i \left( \frac{\eta }{\epsilon }\right) ^{\frac{n-i}{d}}. \end{aligned}$$
(3.2)

Lemma 3.2

For each \(\epsilon \in E(D,\Lambda )\) Eq. (3.2) has the unique solution \(\eta (D,\Lambda ,\epsilon )\), satisfying \(\eta (D,\Lambda ,\epsilon ) > \epsilon \).

Proof

Clearly, the right hand of (3.2) is strictly monotone in \(\eta \), and it tends to \(\infty \) as \(\eta \) tends to \(\infty \). For \(\eta =\epsilon \) the right hand of (3.2) is

$$\begin{aligned} c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i < M(\epsilon , D), \end{aligned}$$

by the definition of \(E(D,\Lambda )\). We conclude that Eq. (3.2) has the unique solution \(\eta (D,\Lambda ,\epsilon ) > \epsilon \). \(\square \)

Now we are ready to state and prove our main result: Theorem 3.2 below. As above, we consider \(C^d\)-smooth mappings \(f:B^n_r\rightarrow {{\mathbb {R}}}^m\), with n, m, d, r fixed. We also fix \(\Lambda =(\lambda _1,\lambda _2, \ldots , \lambda _m)\) as above, and a compact set \(D \subset B^m_1 \subset {{\mathbb {R}}}^m\).

Theorem 3.2

Let f, \(\Lambda \) and D be as above, with \(D \subseteq \Delta (f,\Lambda )\), and with \(E(D,\Lambda ) \ne \emptyset \). Then we have

$$\begin{aligned} R_d(f)\ge \mathrm{max\,}_{\epsilon \in E(D,\Lambda )} \ \ \eta (D,\Lambda ,\epsilon ). \end{aligned}$$

Proof

Since \(D \subseteq \Delta (f,\Lambda )\), then for each \(\epsilon \in E(D,\Lambda )\) we have

$$\begin{aligned} M(\epsilon , D)) \le M(\epsilon , \Delta (f,\Lambda )) \le c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i \left( \frac{R_d(f)}{\epsilon }\right) ^{\frac{n-i}{d}}. \end{aligned}$$
(3.3)

Indeed, by Proposition 3.1 we have, in particular, \(\epsilon \le R_d(f).\) Then (3.3) is provided by the second inequality of Theorem 3.1.

On the other hand, by the definition of \(\eta (D,\Lambda ,\epsilon )\) in Lemma 3.2, we have

$$\begin{aligned} M(\epsilon , D)) = c\sum _{i=0}^m \lambda _0\lambda _1\lambda _2 \ldots \lambda _i\left( \frac{r}{\epsilon }\right) ^i \left( \frac{\eta (D,\Lambda ,\epsilon )}{\epsilon }\right) ^{\frac{n-i}{d}}. \end{aligned}$$
(3.4)

Now comparing (3.3) and (3.4) we conclude, via monotonicity of the right-hand side of these expressions with respect to \(R_d(f)\), that

$$\begin{aligned} R_d(f)\ge \eta (D,\Lambda ,\epsilon ). \end{aligned}$$

Taking maximum of the right-hand side of this inequality with respect to \(\epsilon \in E(D,\Lambda )\), we finally obtain

$$\begin{aligned} R_d(f)\ge \mathrm{max\,}_{\epsilon \in E(D,\Lambda )} \ \ \eta (D,\Lambda ,\epsilon ). \end{aligned}$$

This completes the proof of Theorem 3.2. \(\square \)

3.3 Some examples

The structure of the critical and near-critical values of smooth functions may be pretty complicated (see, e.g., [9, 15] and references therein). The results of the present paper give a new approach to these sets. We consider the problem of understanding the fractal geometry of \(\Delta (f)\), via Theorem 3.2, from the point of view of smooth rigidity, as an important and non-trivial one. In this paper we provide some initial examples, which may be instructive.

In these examples we simplify the general setting of Sect. 3.2. First we restrict the consideration to the case \(m=1\), i.e., to smooth functions \(f:B^n_1\rightarrow {{\mathbb {R}}}\). Second, we consider only critical values of f (and not near-critical ones). That is, we put \(\lambda _1=0\), and consider the sets of critical values \(\Delta (f)\), thus dropping the parameter \(\Lambda \). Theorem 3.1 takes now the form:

$$\begin{aligned} M(\epsilon ,\Delta (f)) \le c(n,d),{} & {} \epsilon \ge R_d(f), \end{aligned}$$
(3.5)
$$\begin{aligned} M(\epsilon ,\Delta (f)) \le c(n,d) \left( \frac{(R_d(f))}{\epsilon }\right) ^{\frac{n}{d}},{} & {} \epsilon \le R_d(f). \end{aligned}$$
(3.6)

Some “explicit” bounds for the constant c(n, d) are given in [15]. However, these bounds involve rather cumbersome expressions. We consider producing more tractable and instructive bounds for the constant c in the present paper, and, in general, in the results of [15], as an important open problem. This problem recently attracted some attention in the field. Below we consider also the case \(n=1\), where \(c(1,d)=d+1\) is explicit.

Now we proceed as follows: we consider a certain close subset \(D \subset [-1,1] \subset {{\mathbb {R}}}\), and find (or estimate) the metric entropy (the covering number) \(M(\epsilon ,D)\) for various \(\epsilon \). On this base we find (or estimate) the set E(D), and then apply Theorem 3.2, in order to provide a lower bound on \(R_d(f)\) for any smooth f with \(D \subset \Delta (f)\).

Assume now that the cardinality of D is strictly greater than c(n, d), and denote, as above, by \({\bar{\epsilon }} = {\bar{\epsilon }}(D,n,d)\) the biggest \(\epsilon \) for which \(M(\epsilon ,D)\ge c(n,d)+1\). Notice, that if the cardinality of D is exactly \(c(n,d)+1\), then \({\bar{\epsilon }}\) is the minimal distance between the points of D.

Next for \({\bar{\epsilon }}(D)\) as above put

$$\begin{aligned} {\bar{\gamma }}(D)= {\bar{\epsilon }}(D)\left[ 1+\frac{1}{c(n,d)}\right] ^{\frac{d}{n}}. \end{aligned}$$
(3.7)

Now we can present our first specific corollary of general Theorem 3.2:

Corollary 3.1

For D and \({\bar{\epsilon }}(D)\) as above, and for each smooth f with

\(D\subseteq \Delta (f))\) we have \(R_d(f)\ge \bar{\gamma }(D)>0\).

Proof

By definition, \({\bar{\epsilon }}(D)\in E\), and thus, according to Theorem 3.2, we can solve uniquely the second equation in (3.5), and denote its solution by \({\bar{\gamma }}(D)\). However, now, in contrast with the general case, we can do it explicitly:

$$\begin{aligned} c(n,d)+1 = M({\bar{\epsilon }}(D),\Delta ) = c(n,d) \left( \frac{(R_d(f))}{\bar{\epsilon }(D)}\right) ^{\frac{n}{d}}, \end{aligned}$$
(3.8)

which gives us

$$\begin{aligned} \left( \frac{(R_d(f))}{{\bar{\epsilon }}(D)}\right) ^{\frac{n}{d}}=\frac{c(n,d)+1}{c(n,d)}=1+\frac{1}{c(n,d)}, \end{aligned}$$

which finally provides

$$\begin{aligned} {\bar{\gamma }}(D)={\bar{\epsilon }}(D) \left[ 1+\frac{1}{c(n,d)}\right] ^{\frac{d}{n}}. \end{aligned}$$

This completes the proof of Corollary 3.1. \(\square \)

Our next result deals with even more restricted situation: we consider smooth functions of one variable. Still we believe that the results provided by Theorem 3.2 in this case are new and instructive. The main advantage of this special case is that the constant \(c(1,d)=d+1\) is explicit and accurate. The following two results can be presented in a “closed form”, not referring to the previous definitions.

Corollary 3.2

Let \(|D|\ge d+1\). Denote by \({\bar{\epsilon }}(D)\) the biggest \(\epsilon \) for which \(M(\epsilon ,D)\ge d+2\). Next put

$$\begin{aligned} {\bar{\gamma }}(D)= {\bar{\epsilon }}(D)\left( 1+\frac{1}{d+1}\right) ^d. \end{aligned}$$
(3.9)

Then for a smooth \(f:B^1 \rightarrow {{\mathbb {R}}}\) with \(D\subseteq \Delta (f)\), we have \(R_d(f)\ge {\bar{\gamma }}(D)>0\).

Proof

This is a special case of Corollary 3.1. \(\square \)

To further illustrate our approach, let’s consider even a more specific case of the situation above:

Corollary 3.3

Let \(|D|\ge 6\). Denote by \({\bar{\epsilon }}(D)\) the biggest \(\epsilon \) for which \(M(\epsilon ,D)\ge 7\). Next put

$$\begin{aligned} {\bar{\gamma }}(D)=\left( \frac{7}{6}\right) ^5{\bar{\epsilon }}(D). \end{aligned}$$
(3.10)

Then for each 5-smooth \(f:B^1 \rightarrow {{\mathbb {R}}}\) with

\(D \subseteq \Delta (f)\) we have

$$\begin{aligned} R_5(f)\ge {\bar{\gamma }}(D)=\left( \frac{7}{6}\right) ^5 {\bar{\epsilon }}(D)>0. \end{aligned}$$

Proof

This is a special case of Corollary 3.1. \(\square \)

Finally, we return to functions \(f:B^n_1\rightarrow {{\mathbb {R}}}, \ n\ge 1,\) and provide another very specific example, which, however, illustrates the power of the lower bounds for the high-order derivatives, provided by Corollary 3.1. We are in a situation of Corollary 3.1, i.e., we consider d-smooth functions \(f: B^n\rightarrow {{\mathbb {R}}}\), and subsets D of their critical values. Lets consider \(D=D_\alpha =\{1, 2^\alpha , 3^\alpha , \ldots , m^\alpha , \ldots \}\). We assume that \(\alpha < 0\), so the points of \(D_\alpha \) always converge to 0. For the differences \(m^\alpha - (m+1)^\alpha \) we get

$$\begin{aligned} m^\alpha -(m+1)^\alpha = m^\alpha \left( 1-\left( 1+\frac{1}{m}\right) ^\alpha \right) =m^\alpha \left( -\frac{\alpha }{m}-...\right) \asymp -\alpha m^{\alpha -1}. \end{aligned}$$

Next we fix a certain sufficiently small value of \(\epsilon >0\), in particular, providing \(M(D_\alpha ,\epsilon )>c(n,d)\). By the above calculation we have \(M(D_\alpha ,\epsilon ) \asymp (\frac{\epsilon }{\alpha })^{\frac{1}{\alpha -1}}.\) Recall that \(\alpha , \ \alpha -1 < 0\).

Finally, by the inequalities above, we get

$$\begin{aligned} M(D_\alpha ,\epsilon ) \asymp \left( \frac{\epsilon }{\alpha }\right) ^{\frac{1}{\alpha -1}} \le c(n,d)\left( \frac{R_d(f)}{\epsilon }\right) ^{\frac{n}{d}}, \end{aligned}$$

or

$$\begin{aligned} R_d(f)\ge \epsilon c(n,d)^{-\frac{d}{n}} \left( \frac{\epsilon }{\alpha }\right) ^{\frac{d}{n(\alpha -1)}}= c(n,d)^{-\frac{d}{n}} \left( \frac{1}{\alpha }\right) ^{\frac{d}{n(\alpha -1)}} \epsilon ^{1+\frac{d}{n(\alpha -1)}}. \end{aligned}$$

What is most important to us in this expression is the dependence of the answer in \(\epsilon \) - in our setting all the other parameters are fixed. Thus we get, for all \(\epsilon \) sufficiently small,

$$\begin{aligned} R_d(f)\ge C \epsilon ^{1+\frac{d}{n(\alpha -1)}}. \end{aligned}$$
(3.11)

The conclusion is that for \(\epsilon \rightarrow 0\) we have, essentially, two quite different possibilities:

1. If the power of \(\epsilon \) is positive, i.e., for \(d\le n(1-\alpha )\), our lower bound (3.11) for \(R_d(f)\) decreases, as \(\epsilon \rightarrow 0\), and this does not exclude \(D_\alpha \) to be a subset of the set of critical values of a \(C^d\)-smooth f. The examples in [15] illustrate this case.

2. If the power of \(\epsilon \) is negative, i.e., for \(d> n(1-\alpha )\), our lower bound (3.11) for \(R_d(f)\) increases, as \(\epsilon \rightarrow 0\). Of course, this excludes a possibility for \(D_\alpha \) to be a subset of the set of critical values of a \(C^d\)-smooth f. The examples in [15] illustrate also this case, but we believe, that the explicit lower bounds for the derivatives of f, given by (3.11) are new.