New York Journal of Mathematics
Volume 27 (2021), 848-880


Johannes Schleischitz

Going-up theorems for simultaneous Diophantine approximation

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Published: June 8, 2021.
Keywords: exponents of Diophantine approximation, parametric geometry of numbers.
Subject: 11J13, 11J83.

We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector ξ=(ξ,ξ2,...,ξN), in various dimensions N. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with Q-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, applied in the framework of parametric geometry of numbers introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality result on polar convex bodies.


The author thanks Yann Bugeaud for fruitful discussions that helped to improve the paper! The author further thanks the referee for pointing out several small inaccuracies.

Author information

Johannes Schleischitz:
Middle East Technical University
Northern Cyprus Campus
Kalkanli, Guzelyurt