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In the Wikipedia page, Siegel's lemma is stated as follows:

Consider the system $$ \begin{cases} \sum_{i=1}^Na_{1i}X_i=0\\ \vdots\\ \sum_{i=1}^Na_{Mi}X_i=0 \end{cases}, $$ where the coefficients $a_{11}, a_{12}, \dots, a_{MN}$ are integers between $-B$ and $B$. The system has a nonzero integer solution $(X_1, X_2, \dots, X_N)$ such that $\max_i \left|X_i\right| \le (NB)^{M/(N-M)}$.

In my understanding, Siegel's lemma is of great importance in Diophantine approximation. However, I haven't been able to find an example of a problem where this formulation of the lemma has been applied to problems in Diophantine approximation. How is this applied in number theory?

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This theorem, as well as algebraic number theoretic extensions, is used to great effect in Ivan Niven’s book: “irrational numbers”, to prove the Gelfond-Schneider theorem (among other things).

I recall it being used in a proof of Baker’s theorem, too.

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There are several works in the general context of approximation of subspaces of Euclidean spaces by lattices, which use Siegel's lemma. To give an example, see the paper Applications of Siegel's Lemma to a system of linear forms and its minimal points. A key result involved is Schmidt's subspace theorem, which has many applications on Diophantine approximation.

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