Angular level

(Joint work with Tobias Kaiser)

The goal of this post is to introduce a rough measure of size for a real domain $U$, based on the level of its boundary function $f_U$. As before, “definable” means “definable in $\Ranexp$”.

Let $\I$ be the set of all infinitely increasing $\,f \in \H$ and set $\bo:= \H_{\gt 0} \setminus \I$. For $u \in \RR$, we let $m_u$ and $p_u$ be the germs at $+\infty$ of the functions $x \mapsto ux$ and $x \mapsto x^u$, respectively.

By a theorem of Marker and Miller, every $f \in \I$ has level, that is, there exist $s,k \in \ZZ$ such that $$\log_k \circ f \sim \log_{k-s},$$ where $h_1 \sim h_2$ if and only if $h_1(t)/h_2(t) \to 1$ as $t \to +\infty$. In this situation, $s$ is unique and called the level of $\,f$, and $\log_l \circ f \sim \log_{l-s}$ for $l \ge k$.

We extend the level to all of $\H_{>0}$ as follows: we set $$\D := \set{1/f:\ f \in \I},$$ and for $f \in \D$, we set the $$\level(f):= \level(1/f).$$ Furthermore, for $\,f \in \H_{\gt 0} \setminus \{\I \cup \D\}$, we set $\level(f):= -\infty$.

Fact (Marker and Miller)
Let $\,f,g \in \H_{\gt 0}$.

  1. If $\,f,g \in \I$ and $f \le g$, then $\level(f) \le \level(g)$.
  2. If $\,f,g \in \bo$ and $f \le g$, then $\level(f) \ge \level(g)$.
  3. If $\,g \in \I$, then $\level(f \circ g) = \level(f) + \level(g)$.
  4. $\level(fg) \le \max\{\level(f),\level(g)\}$; equality holds whenever $\level(f) \ne \level(g)$.

Question
Given $\,f \in \H_{\gt 0}$ and a word $w \in \E^\circ$, how do the levels of $f$ and of $\nu_w(f)$ compare?

Example
If $w = p_r$ or $w = m_r$, for $r>0$, then $\level(f) = \level(\nu_w(f))$.

Things get a little more interesting for $w = \log$:

Lemma
There is an upper cut $\J$ of $\I$ such that, for $\,f \in \H_{\gt 0}$, we have $$\level(\nu_{\log}(f)) \begin{cases} = \max\{0, \level(f) + 1\} &\text{if }\ f \notin \J, \\ \le 0 &\text{if }\ f \in \J. \end{cases}$$

In view of this lemma, we make the following

Definition
The angular level $\alevel:\H_{\gt 0} \into \{-1, 0, \dots \}$ is defined as $$\alevel(f):= \begin{cases} -1 &\text{if }\ f \in \J, \\ \max\{0,\level(f)+1\} &\text{if }\ f \notin \J. \end{cases}$$

Combining the Fact and the Lemma, we obtain:

Proposition
The angular level is decreasing and, for $\,f,g \in \H_{\gt 0}$, we have

  1. for $r\gt 0$, we have $\alevel(\nu_{p_r}(f)) = \alevel(\nu_{m_r}(f)) = \alevel(f)$;
  2. $\alevel(\nu_{\log}(f)) = \alevel(f) + 1$;
  3. if $\,f \in \Hsr$, then $\alevel(\nu_{\exp}(f)) = \alevel(f)\ – 1$.

…which brings us back to the question above. Combining this proposition with (some extra details of) the corollary in this post, we obtain:

Corollary
Let $\,w \in \E^\circ$, and let $f$ be in the domain of $\nu_w$. Then $$\alevel(\nu_w(f)) = \alevel(f)\ – \level(w).$$

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