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Moscow Journal of Combinatorics and Number Theory, vol.11, no.2, pp.125-148, 2022 (Scopus)
© 2022, Mathematical Sciences Publishers. All rights reserved.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 x1 θi,1 + · · · + xn θi,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 ℓ of consecutive best approximation vectors are linearly independent. Our method is based on Siegel’s lemma.