Let me recall some definitions: a set $A \subseteq \mathbb N$ is:

*syndetic*if it intersects every large enough interval,*i.e.*if $\exists \ell \in \mathbb N^* : \forall k \in \mathbb N, A \cap ⟦ k, k+\ell - 1 ⟧ \neq\varnothing$ ;*thick*if it contains arbitrarily long intervals,*i.e.*if $\forall \ell \in \mathbb N^*, \exists k \in \mathbb N : ⟦k, k+\ell-1⟧ \subseteq A$.

I'm interested in bijections $\mathbb N \to \mathbb N$ preserving these two notions. More precisely, I say that a bijection $f : \mathbb N \to \mathbb N$

*preserves thickness*if, for every set $A \subset \mathbb N$, $A$ thick $\implies$ $f[A]$ thick ;*strongly preserves thickness*if, for every set $A \subset \mathbb N$, $A$ thick $\iff$ $f[A]$ thick.

I can define the same notions for syndeticity, but they are at least partly redundant. Indeed, a set $A$ is syndetic iff $\mathbb N \setminus A$ isn't thick. That shows that a bijection strongly preserves thickness iff it strongly preserves syndeticity.

My first question is the following:

Do you have an example of a bijection $f : \mathbb N \to \mathbb N$ which preserves thickness, but doesn't preserve it strongly?

I'm not sure my second question has a satisfying answer, but here it is:

What is a good description of (strongly) thickness-preserving bijections?

Here's a (hopefully correct) very partial answer: if $W(\mathbb N)$ denote the group of bijections $\mathbb N \to \mathbb N$ satisfying $\exists d \in \mathbb N^* : \forall i \in \mathbb N, \left\lvert f(i) - i \right\rvert \leq d$, every $f \in W(\mathbb N)$ strongly preserves thickness, but there are other examples. For instance, if $(a_n)$ is a rapidly growing sequence, I think that the bijection $f$ swapping each $a_{2n}$ with $a_{2n+1}$ strongly preserves thickness, even if $f \not\in W(\mathbb N)$.

**Edit.** I now believe that if $f : \mathbb N \to \mathbb N$ is a bijection, and there exists $d \in \mathbb N^*$ such that the "d-approximate support" $S_d(f) = \left\{ i \in \mathbb N \, \big| \, \left\lvert f(i) - i \right\rvert > d \right\}$ isn't piecewise syndetic, $f$ strongly preserves thickness. Could it be a necessary and sufficient condition?

A set is *piecewise syndetic* iff it can be written as $S \cap T$, where $S$ is syndetic and $T$ thick. It means that, for some $\ell$, it contains arbitrarily long sequences $a_1 < \ldots < a_p$ s.t. $a_{i+1} - a_i \leq \ell$.