Discrete & Computational Geometry

, Volume 39, Issue 1, pp 59-66

First online:

Siegel’s Lemma and Sum-Distinct Sets

  • Iskander AlievAffiliated withSchool of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings Email author 

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Let L(x)=a 1 x 1+a 2 x 2+⋅⋅⋅+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that there exist linearly independent vectors x 1,…,x n−1∈ℤ n such that L(x i )=0, i=1,…,n−1, and
where a=(a 1,a 2,…,a n ) and
$$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$

This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.