On the sharpness of Siegel's Lemma

by · Jul 23, 2016 · 228 views ·

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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