On geometry of simultaneous approximation to three real numbers

Antoine Marnat^ab, Nikolay Moshchevitin<sup>cde

a</sup> Univ. Paris Est Créteil, Univ. Gustave Eiffel, CNRS, LAMA UMR8050, F-94010 Créteil, France
^b Institut für Diskrete Mathematik und Geometrie, TU Wien, Wien, Austria
^c Technische Universität Wien
^d Faculty of Computer Science, HSE University, Pokrovsky boulevard 11, Moscow, Russia 109028
^e Moscow Center of Fundamental and Applied Mathematics

Abstract: Considering simultaneous approximation to three numbers, we study the geometry of the sequence of the best approximations. We provide a sharper lower bound for the ratio of ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.

Key words and phrases: simultaneous approximation, Diophantine exponents, best approximations.

MSC: 11J13

Language: English

DOI: 10.17323/1609-4514-2025-25-1-33-61