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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?

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  • $\begingroup$ It is a trivial consequence of Skolem-Mahler-Lech together with the finiteness theorem on the $k$-variable $S$-unit equation, itself a special case of the Subspace theorem. You will find the argument in chapter 7 of Bombieri and Gubler's book. It is much easier to prove directly that the power sums are piecewise monomials on a suitable arithmetic progression. $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 7, 2016 at 22:56
  • $\begingroup$ (I just noticed you required the $x_i$ to be rational integers. In that case, Skolem-Mahler-Lech is not needed, and your statement is directly a consequence of the theorem stating that all but finitely many solutions to the $k$-variable $S$-unit equation have a proper vanishing subsum. This in turn is an easy consequence of the Subspace theorem, explained in chapter 7 of [BG].) $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 7, 2016 at 23:34
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    $\begingroup$ I agree that appealing to the subspace theorem (and related results) is a bit like using a sledgehammer to crack a nut, and the elementary approach I was alluding to in the OP is along the lines of what you suggest in your 1st comment. On the other hand, I don't see how to use Theorem 7.4.1 in [BG], which is, I think, what you refer to in your 2nd comment, to prove the claim without first showing that, if the contrary were true, there would exist a finite set $P$ of primes s.t. every prime divisor of $c_1x_1^{2n}+\cdots+c_kx_k^{2n}$ is in $P$ for infinitely many $n$, which needs a bit of (...) $\endgroup$
    – Salvo Tringali
    Commented Dec 9, 2016 at 10:05
  • $\begingroup$ (...) work, as far as I can say, and is precisely what I'd like to avoid, since then the conclusion would be more or less straightforward (with or without the subspace theorem). But maybe I'm missing something, as you write of a direct consequence of Evertse, Schlickewei and Schmidt's theorem on the k-variable unit equation (specialized to the rational field), or at least this is how I interpret your words. So thank you in advance for any further comment. $\endgroup$
    – Salvo Tringali
    Commented Dec 9, 2016 at 13:17
  • $\begingroup$ Ah, I see now that I had misread your question. I was assuming precisely the thing you wrote at the end of your upvoted comment above, but it was something different: I was misreading the $\omega(\cdot)$ notation, for the number of distrinct prime factors. Well, OK, then of course you need additional work. You may not appeal to the Subspace theorem. (Actually, I wasn't implying this about the sledgehammer against elementary approaches, and I don't agree with this point of view. The Subspace theorem is just one of many approaches. It just doesn't apply to your $\omega$l question.) $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 9, 2016 at 18:53

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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.

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