Regulators on additive higher Chow groups

As an attempt to understand motives over $k[x]/(x^m)$, we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. R\"ulling on $0$-dimensional cycles. We give an explicit construction of regulator maps on the groups of $1$-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.