Let $U=(u_n)_{n=0}^{\infty}\subseteq\mathbb{C}$ be a sequence enumerated by a *linear homogeneous recurrence relation with constant coefficients*, i.e., there is some $d\geq 1$ and $a_1,\ldots,a_d\in\mathbb{C}$ such that, for $n\geq d$,
$$
u_n=a_1u_{n-1}+\ldots+a_du_{n-d}.
$$

Now, given some $k\geq 1$, a tuple $\bar{c}=(c_1,\ldots,c_k)\in\mathbb{Z}^k$, and some $r\in\mathbb{C}$, define $$ U(\bar{c},r)=\{\bar{n}\in\mathbb{N}^k:c_1u_{n_1}+\ldots+c_ku_{n_k}=r\}. $$ The question I want to ask has to do with the structure of sets of the above form. There has been some well-known work on this type of problem. For example:

**Skolem-Mahler-Lech Theorem**: Set $k=1$, $c_1=1$, and $r=0$. Then
$$
U(1,0)=F\cup P_1\cup\ldots\cup P_t,
$$
where $F$ is finite and each $P_i$ is an infinite arithmetic progression.

On the other hand, results in this area for larger $k$ seem to be less developed. Most things I have found are restricted to $k\leq 3$ (e.g. this paper by Schlickewei and Schmidt). I get the impression that this type of question can be quite difficult for arbitrary recurrence relations and large $k$.

So, I want to know if more can be done in the case that the characteristic polynomial $x^d-a_1x^{d-1}-\ldots-a_{d-1}x-a_d$ of $U$ has no repeated roots. Are there results showing that, for arbitrary $k$, sets of the form $U(\bar{c},r)$ have "nice" structure in this case?