Proving that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is bounded only if $|x_1|=\cdots=|x_k|$ by the Subspace Theorem Let $c_1, \ldots, c_k \in \mathbf N^+$ and $x_1,\ldots,x_k \in \mathbf Z \setminus \{0\}$. It is possible to prove by elementary means that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is a bounded sequence only if $|x_1|=\cdots=|x_k|$. (As usual, $\omega(x)$ is, for every non-zero $x \in \mathbf Z$, the number of distinct prime divisors of $x$, while $\omega(0) := \infty$.)
On the other hand, there seems to be more than a chance that the same conclusion may also come as an easy consequence of the Subspace Theorem (or any of its descendants). This sounds plausible to me, but I don't see how to proceed. So I thought to ask here, with the hope that the question is not completely trivial for the experts in the field and not totally uninteresting for the others.

Q. Can the statement made in the first paragraph of this post be obtained, in a more or less straightforward way, from the Subspace Theorem? If so, is a proof along these lines written down anywhere? 

 A: Just in order to mark this question as answered: The answer is yes. Some details follow.

The basic idea (for some more general conclusion) was generously provided by the anonymous referee of a short note (joint work with Paolo Leonetti) that has been only recently accepted for publication in JNT (*). The key ingredient is Theorem 3 from:

J.-H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), No. 3, 559–601,

which yields, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables (over the additive group of the rationals). 
More precisely, Evertse's theorem implies the existence of a base $\theta \in \mathbf R^+$ such that $\omega(s_n) \gg \text{slog}_\theta(n)$ for infinitely many $n$, where $\text{slog}_\theta$ is a kind of inverse of tetration (the same can be also proved by using only elementary means, but this question was about the Subspace Theorem and its descendants).
(*) I hope this doesn't sound as self-promotion. If it does and you have any suggestion on how to avoid it in cases like this, then I'd appreciate to know.
