, Volume 39, Issue 1-3, pp 59-66

Siegel’s Lemma and Sum-Distinct Sets

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Abstract

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 $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ 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.

The work was partially supported by FWF Austrian Science Fund, Project M821-N12.