Siegel's Lemma is a key tool in transcendental number theory and diophantine approximation. It is a clever application of the Pigeonhole Principle—a ``pure existence argument''—with strikingly powerful consequences.
Suppose we have a homogeneous system of linear equations with integer coefficients: n
equations, N variables with N>n, and every coefficient has absolute value ≤A.
PROBLEM: Give an upper bound to the maximum norm of the smallest nontrivial integer solution.
Sharpest known form of Siegel's Lemma gives the upper bound
(70N−−√A)n/(N−n).
It is easy to show that the factor A cannot be replaced by o(A) (hint: use primes). So, A=1 is the most interesting special case.
Hard Question: Assuming A=1
, can we replace the base N−−√ in the upper bound (N−−√)n/(N−n) with a smaller base o(N−−√) ?
The answer is no: We prove that the sharpest known form of Siegel's Lemma is best possible. We outline the long proof.

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