Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$
of (strictly) positive integers, we can consider subsets
$A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in
$\mathcal S$.

Examples: If $\mathcal S=1+\mathbb N$ we can take
$A=2\mathbb N$. If $\mathcal S=(2n)^{\mathbb N}=\lbrace
1,2n,(2n)^2,\ldots\rbrace$,
we can take for $\mathcal A$ all integers in the
classes of $0,2,4,\ldots,2n-2 \pmod{2n+1}$.

If $\mathcal S=\mathcal F=\lbrace
1,2,3,5,8,13,21,34,\ldots\rbrace$
is the set of Fibonacci numbers, the greedy algorithm
(which adds iteratively the smallest possible element)
starting with $0$ yields
$$A=\lbrace
0,4,10,14,20,24,30,36,40,46,50,56,60,66,72,76,82,\ldots\rbrace
\ .$$
(This sequence is missing in the OEIS, the sequence augmented by $1$ coincides with A314043 up to $56+1=57$ before differing
by $1$ on $60+1=61<62$.)

Denoting by 
$a_1=0<a_2=4<a_3=10<\ldots$ the increasing sequence
underlying $\mathcal A$, the graph of the function
$i\longmapsto \frac{a_i}{i}$ for $i=400,\ldots,12000$
is given by 

[![$i\longmapsto \frac{a_i}{i},i=400\ldots 12000$][1]][1]

This suggests perhaps the existence of a limit-density
of $\mathcal A$ as a subset of $\mathbb N$.


Multiplicities in gap-sizes of the increasing sequence
$a_1,a_2,\ldots$ define
the gap-size polynomial $G(N)=\sum_{i=1}^{N-1}t^{a_{i+1}-a_i}$.
We get
$$G(1000)=354t^4+553t^6+8t^7+9t^9+42t^{10}+t^{11}+11t^{12}+t^{14}
+2t^{15}+9t^{16}+\ldots+t^{28}$$
and
$$G(10000)=3445t^4+5467t^6+40t^7+57t^9+498t^{10}+10t^{11}+121t^{12}+\ldots+t^{54}$$
indicating perhaps also
some sort of limit frequencies for gap-sizes.

A final observation is the rarity of odd elements in
$\mathcal A$: only 97 of the first $10000$ elements
in $\mathcal A$ are odd. (For $n$ an odd number, elements of
$A$ seem to be more ore less equidistributed modulo $n$.)

*Is there an explanation for these (empirical) observations
or are they artefacts?*


  [1]: https://i.sstatic.net/9hpHX.gif