A Deriving an inequality from a paper

Hi, I am studying a paper by Yann Bugeaud:


on page 13 there is an inequality (16) as given below-



which is obtained from -



, on page 12.

How the inequality (16) is derived? I couldn't figure it out. However one of my forum member

tried but it has two problems (problems are marked as "how?"), it is given below-



It is not clear how those two questions would be resolved.

Can any one show the derivation of inequality (16)?




Thanks in Advance.
 
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Are you missing the latex or images for the equations of interest in your post?

I see gaps but no equations.

POSTSCRIPT: I also had to edit your post to make the link more explicit. We sometimes get posts like this that are actually spam with hidden links.
 

fresh_42

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Multiplying it leads to the question whether
$$
\left|\dfrac{\beta^m}{\gamma^n} \cdot \dfrac{\gamma-\delta}{\alpha-\beta}- \dfrac{\delta^n}{\gamma^n} \right| \ll \dfrac{1}{\alpha^{\eta(m-1)}}
$$
given ##\alpha \geq \gamma \geq |\delta|^{1+\eta}\, , \,\alpha \geq |\beta|^{1+\eta}##.

Are there additional relations by the fact that ##(\alpha,\beta)## and ##(\gamma,\delta)## are Lucas pairs?
An idea could be to write the left hand side as a power series in ##\alpha## and estimate the coefficients.
 
Are you missing the latex or images for the equations of interest in your post?

I see gaps but no equations.

POSTSCRIPT: I also had to edit your post to make the link more explicit. We sometimes get posts like this that are actually spam with hidden links.
I used Image from paper.
 
Are there additional relations by the fact that ##(\alpha,\beta)## and ##(\gamma,\delta)## are Lucas pairs?
An idea could be to write the left hand side as a power series in ##\alpha## and estimate the coefficients.
I tried, two other persons tried, see the last image, but failed to get the result, Plz help.
 

fresh_42

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2018 Award
11,153
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I tried, two other persons tried, see the last image, but failed to get the result, Plz help.
Are there additional relations by the fact that ##(\alpha,\beta)## and ##(\gamma,\delta)## are Lucas pairs?
As far as I can see, there is no restriction on ##\eta## except ##0<\eta <\frac{1}{2}.## Let us assume that ##\eta## is close to zero. Then the assertion is that ##0 \ll \dfrac{\alpha (\gamma - \delta)}{\gamma (\alpha -\beta)} \cdot \dfrac{\alpha^{dr}}{\gamma^{ds}}\ll 2##, i.e. the quotient is close to ##1## in this case. This can be written as ##\alpha^m \gamma +\gamma^n \beta \approx \gamma^n\alpha + \alpha^m \delta##.

As this condition only depends on the given Lucas pairs ##u_m(\alpha,\beta) = v_n(\gamma,\delta)##, I assume that both equations are related; which I don't know since I haven't read the paper. If so, then I would start here and look whether we can go backwards to
$$
\left| \dfrac{\alpha (\gamma - \delta)}{\gamma (\alpha -\beta)} \cdot \dfrac{\alpha^{dr}}{\gamma^{ds}} -1 \right| \ll \alpha^{-\eta rd}
$$
 
As this condition only depends on the given Lucas pairs ##u_m(\alpha,\beta) = v_n(\gamma,\delta)##, I assume that both equations are related; which I don't know since I haven't read the paper.
Read page 12 and 13 (at most, page 10, 11) of below pdf file, for all information required for my query, no need to read the whole paper.
http://irma.math.unistra.fr/~bugeaud/travaux/ConfMumbaidef.pdf

The problem is algebraic derivation which I could not figure out.

You wrote-
the assertion is that ## \dfrac{\alpha (\gamma - \delta)}{\gamma (\alpha -\beta)} \cdot \dfrac{\alpha^{dr}}{\gamma^{ds}}\ll 2##,
do you think
##\alpha^{-\eta rd} \approx 2## ? also please see our attempt in the last image of fist post.
anyway, please inform me if find anything, thanks.
 

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