Generalized (𝑡,𝑠)-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series

Abstract

The theory of ( t , s ) (t,s) -sequences leads to powerful constructions of low-discrepancy sequences in an s s -dimensional unit cube. We generalize this theory in order to cover arbitrary sequences constructed by the digital method and, in particular, the Kronecker-type sequences introduced by the second author. We define diophantine approximation constants for formal Laurent series over finite fields and show their connection with the distribution properties of Kronecker-type sequences. The main results include probabilistic theorems on the distribution of sequences constructed by the digital method and on the diophantine approximation character of s s -tuples of formal Laurent series over finite fields.