Under what minimal conditions must the range of a definable, injective function $\omega^{<\omega}\to\Bbb N$ have positive density in the integers?
I'm very inexperienced in such matters but it would appear that even if some countable set bijects into $\Bbb N$, the same set might by some other function only range over into $\{2^n:n\in\Bbb N\}$, in which case its image would not be guaranteed to have positive density in the integers.
So it appears on the fact of it that some further conditions would need to be implied.
But I don't know if this would contradict some countability argument. Perhaps by the nature of $\omega^{<\omega}$, if the length of sequences is truly unlimited, no exponential map exists which could accommodate an unlimited number of limit points and still be countable.
(1). There exists a bijection $g: \omega^{<\omega}\to \Bbb N.$ If $h:\Bbb N\to \Bbb N$ is any injective function then the composite function $hg:\omega^{<\omega}\to \Bbb N$ is injective . Examples: (i). $h=id_{\Bbb N}.$...(ii). $h(n)=2n$ for all $n\in \Bbb N.$...(iii). $h(n)=2^n$ for all $n\in \Bbb N$. ... (iv). Let $S$ be any infinite subset of $\Bbb N$ and let $h:\Bbb N\to S$ be the (unique) order-isomorphism.
(2). For any $S\subset \Bbb N$ and for any $n\in \Bbb N$ let $|n\cap S|$ denote the number of members of $S$ that are less than $n.$
We can construct $S\subset \Bbb N$ such that for every $r\in [0,1]$ there exists a strictly increasing $j_r:\Bbb N\to \Bbb N$ such that $\lim_{n\to \infty} \frac {| j_r(n)\cap S|}{j_r(n)}=r.$
We first construct $S$ such that this holds for every $r\in \Bbb Q\cap [0,1].$ Then for $r\in 0,1] \backslash \Bbb Q$ take a sequence $(q_i)_{i\in \Bbb N}$ of rationals in $[0,1]$ converging to $r.$
Now let $j_r(1)=j_{(q_1)}(m)$ for some $m$ such that $\left|q_1-\frac {|j_r(1)\cap S|}{j_r(1)}\right| <2^{-1}.$
And take $j_r(n+1)= j_{(q_{n+1})}(m)$ for some $m$ such that $j_r(n+1)>j_r(n)$ and such that $\left|q_{n+1}-\frac {|j_r(n+1)\cap S|}{j_r(n+1)}\right|<2^{-n}.$
Then $\lim_{n\to \infty} \frac {|j_r(n)\cap S|}{j_r(n)}=r. $