Abstract
In this paper we deduce a lower bound for the rank of a family of \(p\) vectors in \(\mathbb {R}^k\) (considered as a vector space over the rationals) from the existence of a sequence of linear forms on \(\mathbb {R}^p\), with integer coefficients, which are small at \(k\) points. This is a generalization to vectors of Nesterenko’s linear independence criterion (which corresponds to \(k=1\)), used by Ball–Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski’s theorem on convex bodies.
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The author has been partially supported by Agence Nationale de la Recherche (project HAMOT, ref. ANR 2010 BLAN-0115-01).
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Communicated by A. Constantin.
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Fischler, S. Nesterenko’s linear independence criterion for vectors. Monatsh Math 177, 397–419 (2015). https://doi.org/10.1007/s00605-015-0769-9
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DOI: https://doi.org/10.1007/s00605-015-0769-9