A new exponent of simultaneous rational approximation

Abstract

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

Major changes since the last version, presentation completely rewritten, title changed. To appear in Acta Arithmetica. 12 pages, 3 figures