One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier
type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets
and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier
functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set
Suppose that f: ℝ → ℝ is a given measurable function, periodic by 1. For an α ∈ ℝ put Mnαf(x) = 1/n+1 Σk=0nf(x + kα). Let Γf denote the set of those α’s in (0;1) for which Mnαf(x) converges for almost every x ∈ ℝ. We call Γf the rotation set of f. We proved earlier that from |Γf| > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γf. However, Γf\ℚ can be dense (even c-dense) for non-L1 functions as well. In this paper we show that there are non-L1 functions for which Γf is of Hausdorff dimension one.