Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / Online First articles

Abstract

We introduce a new exponent of simultaneous rational approximation $\widehat{\lambda}_{\min} (\xi ,\eta )$ for pairs of real numbers $\xi ,\eta$, in complement to the classical exponents $\lambda (\xi ,\eta )$ of best approximation, and $\widehat{\lambda} (\xi ,\eta )$ of uniform approximation. It generalizes Fischler’s exponent $\beta _0(\xi )$ in the sense that $\widehat{\lambda}_{\min} (\xi ,\xi ^2) = 1/\beta _0(\xi )$ whenever $\lambda (\xi ,\xi ^2) = 1$. Using parametric geometry of numbers, we provide a complete description of the set of values taken by $(\lambda ,\widehat{\lambda}_{\min} )$ at pairs $(\xi ,\eta )$ with $1,\xi ,\eta$ linearly independent over $\mathbb Q$.