Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:

We say $A$ has infinite difference length, if

**(a)** For every integer $n$ there exist a number $k=2^q$ (for some positive integer $q$) and $a_1,\cdots,a_k\in A$ such that
$$
n=a_1+\cdots+a_{\frac{k}{2}}-(a_{\frac{k}{2}+1}+\cdots+a_k).
$$

Now, denote by $k(n)$ the least $k$ obtained from (a).

**(b)** The set of all $k(n)$, where $n$ runs over all integers, is unbounded above.

For example, if $\gcd\{a,b\}=1$ then $A=\{a,b\}$ has infinite difference length, but not $A=\mathbb{Z}^+$ (it does not have the second condition (b)).

Now, my questions are:

**(1)** Does the set of all Fibonacci numbers have infinite difference length?
(see https://math.stackexchange.com/questions/1989375/representation-of-integers-by-fibonacci-numbers)

**(2)** What about the Euler numbers?

**(3)** Does anybody know some important well-known integer sequences with infinite difference lengths?

**(4)** Did anyone see something like the above property (definition) yet?

Thanks in advance