Dirichlet spectrum for one linear form

Abstract

AbstractFor , we determine the Dirichlet spectrum in with respect to a linear form and the maximum norm as the entire interval [0, 1]. This natural result improves on the recent work of Beresnevich et al. and complements a subsequent paper by the authors where the analogous result was proved for simultaneous approximation. Various generalizations that can be obtained by similar methods as in the latter paper are indicated. We believe that our results are an important step toward resolving the very open analogous problem for a general system of linear forms.