Power and exponential sums for generalized coding systems by a measure theoretic approach

A measure theoretic approach to study the power and the exponential sums for the usual coding system has been developed since the 1990s. In this paper, we first introduce a new coding system, and then give explicit formulas for the power and the exponential sums for the coding system by the measure theoretic approach. An expression for the power sum using the generalized Takagi function will also be given.