Bound to the Number of Solutions of the Thue-Siegel-Roth Theorem

Question

I have to prove for an assignment that if the partial quotients of a number $x$ are $\frac{p_n}{q_n}$, and $$\limsup\frac{\log\log (q_n)\sqrt{\log n}}{n}=\infty$$ then $x$ is transcendental.

I do not want a proof for this claim, as I would like to do my assignment myself

However, I believe that this is intended to be answered based on the Thue-Siegel-Roth theorem, and specifically on a bound on the number of solutions to $$\left|qx-p\right|<\frac{1}{q^{1+\varepsilon}}$$ established by Davenport and Roth (1955) in "Rational approximations to algebraic numbers", Mathematika 2, pp160–167, which was mentioned in a class I missed.

Unfortunately, I have not been able to find this paper online for free, or in my university's library, nor find any other paper quoting the result.

Does anyone happen to know the bound or have access to the paper and can post the result? (or perhaps tell me that I'm wrong, no such magic result exists, and I just didn't think hard enough)

Thanks

EDIT: Here's a link to the suspected paper in Cambridge Journals. I'm sure someone here has access - all you have to do to make me grateful for at least a couple of weeks is open the paper and copy the bound.

Answer 1

"Thue-Siegel-Roth theorem" searching by Google, there are many good references and proofs. See, e.g., Wikipedia, Ishak's paper (pdf), and more. I respect that you want help and not proof, however if you post your proof as an answer, with or without Thue... theorem, I will upvote it, because your problem is interesting.

Answer 2

I found the results, in the paper as you suspected:

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Theorem 1 (Davenport & Roth, 1955) Let $\alpha$ satisfy the irreducible equation $$\alpha^n+a_1\alpha^{n-1}+\ldots+a_n=0$$ where $n\ge 3$ and $a_n\ldots a_n$ are rational integers. Let $$C = 3+\log(a+|\alpha|)+2\log(1+A)$$ where $A=\max\left(|a_1|,\ldots,|a_n|\right)$. Then if $0<\zeta\le\frac{1}{3}$, the number of solutions of $$\left|\alpha-\frac{h}{q}\right| < \frac{1}{2q^{2+\zeta}}$$ in integers $h,q$ with $(h,q)=1$ and $q>0$ is less than $$2\zeta^{-1}\log C + \exp(70n^2 \zeta^{-2})$$

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They actually proceed to show the result you need (which, as you requested, I will not post), using the following corollary:

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Corollary 2 Let $\beta$ satisfy the irreducible equation $$b_0\beta^n+b_1\beta^{n-1}+\ldots+b_n=0$$ where $n\ge 3$ and $b_n\ldots b_n$ are rational integers. Then if $$0<\xi\le\frac{2}{3}, M>\frac{1}{2}$$ the number of solutions of $$\left|\beta-\frac{h}{q}\right| < \frac{1}{2q^{2+\xi}}$$ in integers $h,q$ with $(h,q)=1$ and $q>0$ is less than $$2(2MB)^{2/\xi} + 4\xi^{-1}\log\log(1+2B^n) + \exp(285n^2\xi^{-2})$$ *where $B=\max\left(|b_0|,|b_1|,\ldots,|b_n|\right)$.

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Good luck.