# Baker's theorem for integer combinations of logarithms of integers?

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?

• 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 $n=1$). Jan 21, 2023 at 14:51
• Please feel free to edit if I stated the conditions of the theorem incorrectly, which is quite likely. Jan 21, 2023 at 15:08
• 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? Jan 21, 2023 at 18:38
• I'm interested in rational integers. Jan 21, 2023 at 23:09

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 $$(\alpha_1^s,\dots,\alpha_n^s)$$).
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.