Relative growth rate and contact Banach–Mazur distance

Abstract

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by $d_\mathrm{CBM}$. Explicitly, we consider the following two set-ups, either on a contact manifold $W \times S^1$ where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of $d_\mathrm{CBM}$ are different in these two cases. In the former case the inputs are (contact) star-shaped domains of $W \times S^1$ which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance $d_\mathrm{CBM}$ defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of $d_\mathrm{CBM}$. In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

Appendix

In this section, we provide a reformulation of the relative growth rate $\gamma _{\ge }(\cdot , \cdot )$ on $G_+$, the set of dominants of a semi-group G endowed with a bi-invariant partial order $\ge$. In fact, we give a reformulation of $\rho _{\ge }^+(\cdot , \cdot )$ on $G_+$ defined by (11). By definition, $\rho _{\ge }^+(a,b)$ could be negative, so in what follows we only discuss the case where $\rho _{\ge }^+(a,b)>0$ for $a, b \in G_+$. The case where $\rho _{\ge }^+(a,b)<0$ can be investigated in a symmetric way.

For any $a, b \in G_+$, we call the pair $(k,l) \in {\mathbb {N}}^2$ an (a, b)-ordering pair if $a^k \ge b^l$. Denote by ${\mathcal {O}}_{(a,b)}$ the set of (a, b)-ordering pairs. Recall that

$$\begin{aligned} \rho ^{+}_{\ge } (a,b) = \inf \left\{ \frac{k}{l} \in {\mathbb {Q}}_+ \,\bigg | \, (k,l) \in {\mathcal {O}}_{(a,b)} \right\} . \end{aligned}$$

Denote by ${\mathcal {P}}_{(a,b)}$ the subset of ${\mathcal {O}}_{(a,b)}$ which consists of all the (a, b)-ordering pairs $(p,q) \in {\mathcal {O}}_{(a,b)}$ such that both p and q are prime numbers. Then we have the following property.

Proposition 4

For any $a,b \in G_+$, we have

$$\begin{aligned} \rho ^{+}_{\ge }(a,b) =\inf \left\{ \frac{p}{q} \in {\mathbb {Q}}_+\,\bigg | \, (p,q) \in {\mathcal {P}}_{(a,b)} \right\} . \end{aligned}$$

(38)

Proof

Denote the left-hand side of the Eq. (38) by L and the right-hand side of the Eq. (38) by R. Since ${\mathcal {P}}_{(a,b)} \subset {\mathcal {O}}_{(a,b)}$, it is obvious that $L\le R$. For the other direction, by definition $L =\rho _{\ge }^+(a,b)$ implies there exists a sequence of pairs $\{(k_n, l_n)\}_{n \in {\mathbb {N}}}$ in ${\mathcal {O}}_{(a,b)}$ such that $\lim _{n \rightarrow \infty } \frac{k_n}{l_n} =L$. Passing to a subsequence, we can assume that $l_n \rightarrow \infty$. In particular, if $L = \frac{k}{l}$ for some $(k, l) \in {\mathcal {O}}_{(a,b)}$, then consider $(k_n, l_n) = (nk, nl)$. Since $L > 0$, we know that $k_n \rightarrow \infty$. Pick $\alpha = 0.6$, then Baker–Harman–Pintz’s theorem in [3] implies that when $n>>1$ there exists a prime number $q_n \in [l_n - l_n^{\alpha }, l_n]$. In particular,

$$\begin{aligned} q_n \le l_n \,\,\,\,\text{ and }\,\,\,\, \lim _{n \rightarrow \infty } \frac{q_n - l_n}{l_n} = 0. \end{aligned}$$

(39)

Similarly, there exists a prime number $p_n \in [k_n, k_n + k_n^{\alpha }]$. In particular,

$$\begin{aligned} p_n \ge k_n \,\,\,\,\text{ and }\,\,\,\, \lim _{n \rightarrow \infty } \frac{p_n - k_n}{k_n} = 0. \end{aligned}$$

(40)

Each $(k_n, l_n)$ belongs to ${\mathcal {O}}_{(a,b)}$, so $a^{p_n} \ge a^{k_n} \ge b^{l_n} \ge b^{q_n}$. Therefore, $(p_n, q_n) \in {\mathcal {P}}_{(a, b)}$. Moreover, due to relations (39) and (40),

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{p_n}{q_n} = \lim _{n \rightarrow \infty } \frac{p_n - k_n + k_n}{q_n - l_n + l_n} =\lim _{n \rightarrow \infty } \frac{\left( \frac{p_n - k_n}{k_n} +1 \right) k_n}{\left( \frac{q_n - l_n}{l_n} +1 \right) l_n} = \lim _{n\rightarrow \infty } \frac{k_n}{l_n} = L. \end{aligned}$$

This implies that $R \le L$, which is the desired conclusion. $\square$

Remark 16

The only place in the proof of Proposition 4 where we use the hypothesis that $L = \rho _{\ge }^+(a, b)> 0$ is to guarantee that the sequence $\{k_n\}_{n \in {\mathbb {N}}}$ tends to infinity (so that we can apply Baker–Harman–Pintz’s theorem from [3] to obtain an (a, b)-ordering pair in ${\mathcal {P}}_{(a,b)}$).