We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, Hölder regularity, and fractional differentiability. Our first main result shows that for every $s \in (1,s]$, the set of continuous functions on $[0, 1]$ whose graph has Hausdorff and box dimensions equal to s is strongly $\mathfrak{c}$-algebrable, thereby tackling an open question by Bonilla et al., and complementing recent findings by Liu et. al and Carmona et al. We then extend the analysis to Hölder spaces: although the pointwise Hölder exponent of a generic function in $C^α[0, 1]$ is constant, we prove that the collection of functions realizing this behavior is $\mathfrak{c}$-lineable but cannot form an algebra. Nevertheless, we construct strongly $\mathfrak{c}$-algebrable families of functions that exhibit Hölder exponent $α$ outside a set of Hausdorff dimension zero. Finally, as a consequence of the relation between strongly monoHölder functions and fractional differentiability, we analyze the strong $\mathfrak{c}$-algebrability of nowhere (Riemann-Liouville) fractional differentiable functions.
Weierstrass' function is defined by
with 0 < a < 1 and ab ≥ 1, and it is a classical example in real analysis of a function that is continuous everywhere but differentiable nowhere. Besides, the graph of W a,b and some of its variations have been studied from a geometric point of view as fractal curves in the plane, see e.g. [9,10,31,32,34,46,48] and references therein. For notation purposes, we will not distinguish between a function f and its graph throughout this paper. While the box dimension of the graph of a function can be easily computed thanks to its Hölder regularity [22], the problem of the computation of the Hausdorff dimension is more complicated to tackle, and it is only a few years ago that the conjecture related to the Hausdorff dimension of the graph of the Weierstrass’ function has been solved [42,47]. In particular, (1) dim
where dim H (•) and dim B (•) denote the Hausdorff and box dimension, respectively. It may be considered as another shock that these pathological properties are not rare from several mathematical points of view. For instance, the residuality of the set of nowhere differentiable functions in the space of continuous function has been obtained in 1931 as a nice application of the Baire category theorem by Banach and Mazurkiewicz independently. In 1994, Hunt [33] extended the latter result to the generic setting of prevalence, a concept introduced in order to generalize the notion of Lebesgue almost everywhere to infinite dimensional spaces [18,35]. Let us remark that researches related to the residuality and prevalence of the set of functions having a graph with a prescribed box or Hausdorff dimension have also been undertaken, see e.g. [13,19,27,36].
Another notion that can show a pathological property is not algebraically rare is lineability. The theory of lineability has provided a number of concepts in order to quantify the existence of linear or algebraic structures inside a-not necessarily linear-set. The term lineability was coined by V. I. Gurariy [3] and is introduced in the Ph.D. Dissertation of the fourth author [49]. It is a well-established line of research in mathematics now. For a deeper understanding, we refer the interested reader to [4]. It is important to mention that, although lineability was first established by Gurariy, lineability results can be traced back to a paper of 1941 by Levin and Milman [44].
The algebraic structure of the set of nowhere differentiable functions in the space of continuous function has also been deeply investigated using this new notion, see e.g. [12,25,29,40]. Also, very recently several authors have analyzed the existence of large spaces with additional properties within the set of continuous nowhere differentiable functions [1,24].
Working in this framework, Bonilla, Muñoz-Fernández, Prado-Bassas and the fourth author in [15] showed that for a given s ∈ (1, 2], the set of continuous functions whose graph has Hausdorff and box dimension equal to s is c-lineable, where c denotes the continuum, and the existing subspace can be chosen to be dense in C[0, 1], the Banach space of continuous functions on [0, 1] with the supreum norm (for definitions see Section 2). They raised the following two questions ([15, Questions 2.11 and 3.4], respectively) of knowing whether besides asking for vector subspaces, one could also study other structures, such as algebras.
Question 1.1. Given s ∈ (1,2], is it possible to obtain the algebrability of the set of functions
, is it possible to obtain maximal-dense-lineability (or algebrability) of the set of functions f ∈ C[0, 1] whose graph have box dimension s everywhere in [0, 1]?
In [45, Theorem 2.7], Liu et. al answered Question 1.2 in the affirmative. Observe that in terms of algebrability, Question 1.1 is very general and does not require any specific conditions on the algebra. In this regard, Liu et. al in [45,Theorem 2.7] showed that the set of functions f ∈ C[0, 1] with dim H (f ) = s ∈ (1, 2] is maximal-dense-lineable and also dense-algebrable.
In this paper, after providing the necessary notions and some primary results in Section 2, we obtain an affirmative answer to Question 1.1 by showing that the the set of functions
is strongly c-algebrable (this is an immediate consequence of Theorem 3.2 in Section 3). In Section 4, we analyze these types of problems in the context of Hölder spaces, which complement the findings of Liu et. al in [45,Section 3]. In fact, we recall classical results concerning the Hölder pointwise regularity of a generic function in a given Hölder space, and then prove that the same holds with the notion of lineability. We show that this generic behavior cannot hold on an algebra. Nonetheless, we obtain a positive (but weaker) result if one allows the Hölder exponent to differ on a set of Hausdorff dimension 0. The results of this section are obtained using wavelets and Schauder bases, and in particular the characterization of the Hölder regularity using
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