Iterated Cesàro averages, frequencies of digits, and Baire category

On Higher Order Irregular Sets

“…For example, Albeverio, Pratsiovytyi and Torbin [1], Hyde et al [12] and Olsen [21] proved that some kinds of irregular sets associated with integer expansion are residual, and the result in [21] was generalized to iterated function systems by Baek and Olsen [2]. Very recently, Madritsch [19] discussed the set of extremely non-normal points associated with Markov partition from the topological point of view and his result generalized the results in [12] and [21]. Moreover, Li and Wu [15] proved that the set of divergence points of self-similar measure with the open set condition is either residual or empty.…”

“…Unfortunately, the approach from [3] cannot be applied in this higher order case. To prove Theorem 1.1, we will use an approach inspired by the idea in [12].…”

“…Inspired by the idea in [12], we define the property P as follows. Let g 1 (x) = 2 x and g m (x) = g 1 (g m−1 (x)) for m ≥ 2.…”

Points with maximal Birkhoff average oscillation

“…Obviously, Theorem 1.1 follows readily from Theorem 1.3 since X max ⊂ X ϕ,f . In the setting of frequencies of N -adic digits, several authors have studied "points of maximal oscillation" and obtained results similar to Theorem 1.3, see [2], [14], [22], [33].…”

“…Recently, some results show that certain irregular sets can also be large from the topological point of view. For example, Volkmann [33],Šalát [29], Albeverio, Pratsiovytyi and Torbin [1], Hyde et al [14] and Olsen [22] proved that some kinds of irregular sets associated with integer expansion are residual. Baek and Olsen [2] discussed the set of extremely non-normal points of a self-similar set from the topological point of view.…”