A new exponent of simultaneous rational approximation

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 $\mathbf{Q}$.