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.



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




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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:



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


