The signed error-sum function of tent-map expansions

Abstract

In this paper, we study the signed error-sum function of tent map base series introduced by Dai and Tang in 2011. We show that the function is discontinuous at every dyadic rational in the open unit interval, although it was asserted to be continuous. We determine its values at these points and analyze its range, differentiability, and variation. We also obtain sharp covering estimates for its graph and calculate the Hausdorff, box-counting, and covering dimensions of the graph, which are $1$, $1$, and $0$, respectively. Moreover, we show that the signed error sum agrees with Lévy's function away from dyadic rationals and that the unsigned error sum is the Takagi function.