No abstract

We provide a lower bound for the ratio between the ordinary and uniform exponents of both simultaneous Diophantine approximation to n real numbers and Diophantine approximation for one linear form in n variables. This question was first considered in the 1950s by Jarník who solved the problem for two real numbers and established certain bounds in higher dimension. Recently different authors reconsidered the question, solving the problem in dimension three with different methods. Considering a new concept of parametric geometry of numbers, Schmidt and Summerer conjectured that the optimal lower bound is reached at regular systems. It follows from a remarkable result of Roy that this lower bound is then optimal. In the present paper, we give a proof of this conjecture by Schmidt and Summerer.

We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.

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It is well known that in one dimension the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist uncountably many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement concerning well approximable sets.