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Baker's theorem in transcendental number theory states that $$ \left|\beta_0 + \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C} $$ where

  • $\beta_0, \ldots, \beta_n$ are algebraic numbers, not all zero,
  • $\alpha_1, \ldots, \alpha_n$ are multiplicatively independent algebraic numbers,
  • $H$ is the maximum of the heights of $\beta_i$, and
  • $C$ is an effectively computable constant depending on $n$, $\alpha_i$, and the maximum of the degrees of $\beta_i$.

Is there a simple proof for a special case of Baker's theorem where $\beta_0 = 0$, the $\beta_i$ are rational integers, and the $\alpha_i$ are positive rational integers?

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    $\begingroup$ I think you actually want your $\alpha_i$ to be linearly independent in $\overline{\mathbb{Q}}^{\times} \otimes \mathbb{Q}$ (otherwise in your case you would have $n=1$). $\endgroup$
    – Aphelli
    Jan 21, 2023 at 14:51
  • $\begingroup$ Please feel free to edit if I stated the conditions of the theorem incorrectly, which is quite likely. $\endgroup$
    – Dave R
    Jan 21, 2023 at 15:08
  • $\begingroup$ Just checking: are you asking about the case where $\alpha_i \in \mathbb{N}^{\star}$ and $\beta_i \in \mathbb{Z}$, or did “integers” refer to algebraic integers in general? $\endgroup$
    – Aphelli
    Jan 21, 2023 at 18:38
  • $\begingroup$ I'm interested in rational integers. $\endgroup$
    – Dave R
    Jan 21, 2023 at 23:09

1 Answer 1

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The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's Diophantine approximation on linear groups gives a simple proof for the homogeneous case $\beta_0=0$. Some technicalities (like using Fel'dman integer-valued polynomials) are not necessary if you don't care about strong estimates.

The homogeneous case is slightly simpler because one can work with just $G_m^n$ or (dually) with $G_a^n$. However, the basic structure of the proof is the same:

  1. Using group structure, consider several points (working with $G_m^n$, the points are $(\alpha_1^s,\dots,\alpha_n^s)$).
  2. construct an auxiliary function/polynomial either using Dirichlet principle (in the form of Siegel's lemma) or using interpolating determinant
  3. Show that this polynomial with some derivatives is small (using either Schwarz's lemma or Hermite's formula) at those points.
  4. Show that it can't vanish at all those points.
  5. Show that if the polynomial or its derivative is small at one of those points, the value must be zero. Assumptions, that $\beta_i$ and $\alpha_i$ are integers simplify the last step - instead of Liouville's inequalities, you are just using directly that if $x$ is an integer with $|x|<1$ then $x=0$. However, the other steps are the same.

Baker's theorem can also be proved using the Schnieder-Lang criterion, but I don’t know if that method will provide estimates sufficient for your needs.

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