2012/09/08 by Nikolay Moshchevitin, Nikolay G. Moshchevitin, Moshchevitin, Nikolay G.
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT #msc:11J13
paper · pdf · doi:10.48550/arxiv.1209.1697
15 pages
arxiv created 2012/09/08 · arxiv updated 2012/09/11
We improve on Jarn'ık's inequality between uniform Diophantine exponent α and ordinary Diophantine exponent β for a system of n≥ 2 real linear forms in two integer variables. Jarn'ık (1949, 1954) proved that β≥ α(α-1). In the present paper we give a better bound in the case α>1. We prove that β≥ 1/2(α2-α+1+√((α2-α+1)2 +4α2(α-1))) if 1≤ α≤ 2 1/2(α2-1+√((α2-1)2+4α(α-1))) if α≥ 2