Vol. 11, No. 2, 2022

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Applications of Siegel's lemma to a system of linear forms and its minimal points

Johannes Schleischitz

Vol. 11 (2022), No. 2, 125–148

Consider a real matrix Θ consisting of rows (𝜃i,1,,𝜃i,n) for 1 i m. The problem of making the system of linear forms x1𝜃i,1 + + xn𝜃i,n yi for integers xj, yi small naturally induces an ordinary and a uniform exponent of approximation, denoted by w(Θ) and ŵ(Θ) respectively. For m = 1, a sharp lower bound for the ratio w(Θ)ŵ(Θ) was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to Θ. Our bound applies to general m > 1, but is probably not optimal in this case. Thereby we also complement a similar conditional result of Moshchevitin, who imposed a different assumption on the best approximations. Our hypothesis is satisfied in particular for m = 1, n = 2 and thereby unconditionally confirms a previous observation of Jarník. We formulate our results in a very general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number of consecutive best approximation vectors are linearly independent. Our method is based on Siegel’s lemma.

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linear forms, best approximations, degenerate dimension phenomenon
Mathematical Subject Classification 2010
Primary: 11J13
Secondary: 11J82
Received: 5 September 2019
Revised: 22 March 2022
Accepted: 5 April 2022
Published: 13 August 2022
Johannes Schleischitz
Middle East Technical University
Northern Cyprus Campus
Northern Cyprus