Hausdorff and Box dimensions of continuous functions and lineability

HAUSDORFF AND BOX DIMENSIONS OF CONTINUOUS FUNCTIONS AND LINEABILITY A. BONILLA, G.A. MU˜ NOZ-FERN´ ANDEZ, J.A. PRADO-BASSAS, AND J.B. SEOANE-SEP´ ULVEDA Abstract. Given s∈(1,2], we study (among other questions) the algebraic genericity of the set of continuous functions f: [0,1] →R whose graph has Hausdorff (or Box) dimension exactly s. 1. Introduction and preliminaries It is a classical conjecture of Benoˆıt B. Mandelbrot from 1977 (see [17]) that the Hausdorff dimension of the graph of the Weierstrass function (1.1) Wλ,b(x) = ∞ X n=0 λncos(2πbnx)x∈R is equal to D= 2 + log λ log b, where b∈N, 0 <λ<1< b and λb > 1. Many authors have worked in this problem for over 30 years. This conjecture has been proved recently by W. Shen ([21], see also [16]). In fact, due to the periodicity of the Weierstrass function, its Hausdorff dimension is the same if it is restricted to [0,1]. Given a function f:A⊂R→R, by graph(f) we shall denote the graph of this function, that is, graph(f) = {(x, f(x)), x ∈A} ⊂ R2. If B⊂A, by graphB(f) we mean the set {(x, f(x)), x ∈B}, that is, the part of graph(f) supported on B. In order to make this article self-contained, we now include the definitions of Hausdorff and Box dimensions. Given a set F⊂Rn, the diameter of Fis |F|:= sup{|x−y|:x, y ∈F}. Given δ > 0, a δ-cover of Fis a countable collection (Ui)iof subsets of Rn such that F⊂SiUiand |Ui|< δ for all i. 2010 Mathematics Subject Classification. 11K55, 15A03. Key words and phrases. lineability, spaceability, algebrability, Hausdorff dimension, Box dimension. A. Bonilla was supported by MINECO MTM2016-75963-P. J.A. Prado-Bassas was supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 Grant P08-FQM-03543 and by MINECO Grant MTM2015-65242-C2-1-P. G.A. Mu˜noz- Fern´andez and J.B. Seoane-Sep´ulveda were supported by Grant MTM2015-65825-P. 1 2 BONILLA, MU˜ NOZ, PRADO, AND SEOANE Figure 1. A sketch of the famous Weierstrass’ Monster (for λ= 1/2 and b= 3), of Hausdorff dimension equals to 2 − log(2) log(3) ≈1.3690702464. For any subset Mof RNand any δ > 0, let Hs δ(M) := inf (∞ X i=1 |Ui|s: (Ui)iis a δ−cover of M), and write Hs(M) = lim δ→0Hs δ(M). The Hausdorff dimension of Mis dimH(M) := inf{s≥0 : Hs(M)=0}= sup{s≥0 : Hs(M)=+∞}. Furthermore, given any non-empty subset Mof Rn, let Nδ(M) denote the smallest number of sets, of diameter at most δ, needed to cover M. Then the lower and upper box dimensions of Mare dimB(M) := lim inf δ→0 log Nδ(M) −log δand dimB(M) := lim sup δ→0 log Nδ(M) −log δ. When these numbers are equal, the common value will be the box dimension of M, that is, dimB(M) := lim δ→0 log Nδ(M) −log δ. For more details on the above definitions and their properties, see [13, Chapters 2–3]. Notice that, in particular, dimH(M)≤dimB(M)≤dimB(M) for any subset Mof Rn. HAUSDORFF AND BOX DIMENSIONS VIA LINEABILITY 3 As usual, we endow the space C[0,1] of continuous functions [0,1] →R with the maximum norm. Observe that it is an infinite dimensional separable Banach space, hence its dimension is exactly c, where the last symbol stands for the continuum (as usual). In this paper, we are interested in the following type of question: Consider the subset Hof continuous functions f: [0,1] →R with dimH(graph(f)) = s. Is there an infinite dimensional subspace X⊂ H∪{0}every non-zero element of which has Hausdorff dimension s? What about the same problem replacing Hausdorff dimension by Box dimension? This kind of questions (related ones are dealt with in [1, 9]) fall under the area of research nowadays known as lineability and spaceability. Let us provide a brief summary of the definitions from this area and that we shall need along this paper. Assume that Xis a vector space and αis a cardinal number. Then a subset A⊂Xis said to be: •lineable if there is an infinite dimensional vector space Msuch that M\{0} ⊂ A. •α-lineable if there exists a vector space Mwith dim(M) = αand M\{0} ⊂ A. •maximal-lineable if it is dim(X)-lineable. If, in addition, Xis a topological vector space, then the subset Ais said to be: •spaceable in Xwhenever there is a closed infinite-dimensional vector subspace Mof Xsuch that M\{0} ⊂ A. •dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\{0} ⊂ A. •α-dense-lineable in Xwhenever there is a dense vector subspace M of Xwith dim(M) = αand M\{0} ⊂ A. •maximal-dense-lineable in Xwhenever there is a dense vector subspace Mof Xwith dim(M) = dim(X) and M\{0} ⊂ A. And, provided that Xis a vector space contained in some (linear) algebra, then Ais called: •algebrable if there is an algebra Mso that M\{0} ⊂ Aand Mis infinitely generated, that is, the cardinality of any system of generators of Mis infinite. •strongly α-algebrable if there exists an α-generated free algebra M with M\{0} ⊂ A. Recall that if Xis contained in a commutative algebra, then a set B⊂Xis a generating set of some free algebra contained in Aif and only if for any N∈N, any nonzero polynomial Pin Nvariables without constant term and any distinct f1, . . . , fN∈B, we have P(f1, . . . , fN)∈A\{0}. 4 BONILLA, MU˜ NOZ, PRADO, AND SEOANE For an account of results on lineability, the reader is referred to the survey [10], the monograph [2], and the works [4, 11,12, 14, 20] in order to see the many fields in which this theory has made an impact so far. Here, Ishall denote a non-trivial interval of R. The following definition and proposition can be found in [6] or [13]. Definition 1.1. Let f:I→R. (1) We say that fis H¨older continuous with exponent α > 0if there exist c, δ > 0such that |f(x)−f(y)| ≤ c|x−y|α for every x, y ∈Isuch that |x−y|< δ. (2) We say that fsatisfies the lower H¨older condition, with exponent β > 0, if there exist ˜ δ, ˜c > 0such that sup x∈J f(x)−inf x∈Jf(x)≥˜c|J|β for every interval J⊂Iwith |J|<˜ δ. Proposition 1.2. If f:I→Ris H¨older continuous with exponent α∈ (0,1], then dimB(graph(f)) ≤2−α. If a continuous function f:I→Rsatisfies the lower H¨older continuous condition with exponent β∈(0,1], then dimB(graph(f)) ≥2−β. This paper is arranged in two main sections. Section 2 shall deal with continuous functions with prescribed Box and Hausdorff dimension of the graph and the problem of lineability of this class of functions. Section 3 will cover the same problem but, in this case, in relation to the Box dimension everywhere. Open questions and directions of research shall also be provided within this topic. 2. On continuous functions with prescribed Box and Hausdorff dimension of their graphs Let us, first, recall a series of results that will be useful for the purpose of this paper. The first one is a direct consequence of the basic properties of the Hausdorff measure. Lemma 2.1. The Hausdorff dimension of a subset of R2is invariant under translations and dilations. Lemma 2.2. Given a number s∈[1,2], a compact interval [a, b]and two real numbers v, w, there exists a continuous function f: [a, b]→Rsuch that f(a) = v,f(b) = wand dimH(graph(f)) = s. HAUSDORFF AND BOX DIMENSIONS VIA LINEABILITY 5 Proof. The case s= 1 is straightforward. We know that the Hausdorff dimension of the graph of the Weierstrass function on [0,1] is D= 2 + log λ log b. Therefore, given s∈(1,2), the functions Fs:= Wλ,b have a graph with Hausdorff dimension sfor a suitable choice of λand b. If s= 2, then the function F2(x) = ∞ X n=1 n−√nsin(nn2πx) has graph of Hausdorff dimension 2 (see [5] and Figure 2). Figure 2. A sketch of function F2from the proof of Lemma 2.2. Here, graph(F2) has Hausdorff dimension 2. By Lemma 2.1, we can translate and dilate, for any s∈(1,2], the function Fsin order to obtain ˜ Fs: [a, b]→Rwith dimH(graph(˜ Fs)) = s. Finally take into account that, according to [18, Theorem 1], the Hausdorff dimension of the graph of a function does not change by adding a Lipschitz function, so that we can add a suitable affine function in order to fulfill the property on the extremes of the interval.  Theorem 2.3. Given s∈(1,2], the set of continuous functions f: [0,1] → Rwith dimH(graph(f)) = sis spaceable. Proof. Fixed s∈(1,2] we split [0,1] into infinitely many intervals with disjoint interior as follows: [0,1] = {1}∪ · [ n∈N In, where In=h1−1 n,1−1 n+1ifor n∈N. Now, by Lemma 2.2, for every n∈Nthere exists a continuous function fn: [0,1] →Rsuch that: 6 BONILLA, MU˜ NOZ, PRADO, AND SEOANE (i) fn(1 −1 n) = 0, (ii) fn(1 −1 n+1) = 0, (iii) fnis null on [0,1] \In, (iv) kfnk∞<1/n, and (v) dimH(graph(fn)) = s. It is clear that the family {fn:n∈N}is linearly independent (since the interior of their corresponding supports are pairwise disjoint). Next, let M be the closed linear span of {fn:n∈N}. Again due the disjointness of the interior of the supports, every nonzero member of Mis a finite or infinite sum of fn’s that, on each In, is a multiple of the corresponding fn. Therefore its graph has Hausdorff dimension s. By (i) and (ii), any nonzero member of Mis obviously continuous on [0,1) (observe that, possibly, the supports of the functions fn’s included in the member of Mmay accumulate at 1). Finally, condition (iv) and the fact that fn(1) = 0, guarantee the continuity at 1.  Lemma 2.4. Let s∈(1,2], the set of functions f∈ C[0,1] with dimH(graph[0,1](f)) = dimB(graph[0,1](f)) = s. is c-lineable. Proof. Let s∈(1,2) and M=span{fa: 0 ≤a < 1}where fa(x) =        Wλ,b x−a 1−a−Wλ,b(0) if x∈[a, 1], 0 if x < a, with parameters b≥2 and 1/b < λ < 1 chosen such that dimH(graph[0,1](fa)) = s. To show the linear independence, we shall proceed by induction. First, we show that for any two functions {fa1, fa2}are linearly independent with a1< a2. Suppose that λ1fa1+λ2fa2= 0. Then for any x∈(a1, a2), λ1fa1(x) + λ2fa2(x) = λ1fa1(x) = 0, thus λ1= 0 and hence λ2= 0. Now suppose that any set {fai: 0 ≤a1<···al<1}with l≤N−1 is linearly independent. Let us suppose that λ1fa1+···+λNfaN= 0, where 0≤a1< a2<···aN<1. Thus, for any x∈(a1, a2) λ1fa1(x) + ···+λNfaN(x) = λ1fa1(x)=0. Thus, λ1= 0 and by inductional assumption, hence λ2=··· =λN= 0. Now, we will show that for any non-trivial combination ( λi6= 0 for 1≤i≤N) with 0 ≤a1< a2<···aN<1 dimH(graph[0,1](λ1fa1+···+λNfaN)) =dimB(graph[0,1](λ1fa1+···+λNfaN)) = s. HAUSDORFF AND BOX DIMENSIONS VIA LINEABILITY 7 The upper bound of dimB(graph[0,1](λ1fa1+···+λNfaN)) ≤sby the H¨older continuity of λ1fa1+···+λNfaN. On the other hand, dimH(graph[0,1](λ1fa1+···+λNfaN)≥ ≥dimH(graph[a1,a2](λ1fa1+···+λNfaN)) = =dimH(graph[a1,a2](λ1fa1)) = =dimH(graph[a1,a2](fa1)) = s. Now, since dimH(M)≤dimB(M)≤dimB(M), we obtain the result. For s= 2, we change Wλ,b by F2(x) = ∞ X n=1 n−√nsin(nn2πx).  We shall now need an auxiliary, general result about lineability: Let Xbe a vector space and A, B be two subsets of X. According to [3], we say that Ais stronger than Bwhenever A+B⊂A. The following lemma (of which many variants have been proved) can be found in [2, 3, 10] and references therein. Lemma 2.5. Assume that Xis a metrizable topological vector space. Let A⊂Xbe (maximal) lineable. Suppose that there exists a dense-lineable subset B⊂Xsuch that Ais stronger than Band A∩B=∅. Then Ais (maximal) dense-lineable in X. Theorem 2.6. Given s∈(1,2], the set of functions f∈ C[0,1] with dimH(graph(f)) = dimB(graph(f)) = s is maximal dense-lineable in C[0,1]. Proof. Denote A(s) := {f∈ C[0,1] : dimH(graph(f)) = dimB(graph(f)) = s}. It is straightforward that the set Pof all polygonal lines (i.e., continuous piecewise affine functions) is a dense linear subset of C[0,1]. In addition, we have that for any s∈(1,2], A(s) is stronger than Pbecause dimH(graph(f+h)) = dimH(graph(f)) and dimB(graph(f+h)) = dimB(graph(f)) for every Lipschitz functions h(see, [18, Theorem 1], [5]). Finally, taking into account that A(s) is c-lineable by Lemma 2.4, a direct application of Lemma 2.5 helps us to conclude the proof.  8 BONILLA, MU˜ NOZ, PRADO, AND SEOANE All previous results can be extended to higher dimensions just by using properties regarding the Hausdorff dimension and cartesian product of sets. It is well known that given any pair of subsets E⊂Rnand F⊂Rm, we only can assert that dimH(E×F)≥dimH(E) + dimH(F). But if one of the subsets is regular enough, the equality holds (see [13, Section 7.1]). Lemma 2.7. For any pair of subsets E⊂Rnand F⊂Rmwith dimH(F) = dimB(F), then dimH(E×F) = dimH(E) + dimH(F). This result is false in general, if we change Hausdorff-dimension by any type of Box-dimension. Theorem 2.8. Given N∈Nand S∈(N, N + 1], the set of functions F: [0,1]N→Rsuch that its graph, graph(F) = {(x1, . . . , xN, F(x1, . . . , xN)) ∈RN+1, xi∈[0,1],1≤i≤N}, has Hausdorff dimension exactly S, is spaceable and maximal dense-lineable in C([0,1]N). Proof. Fix a f∈ C([0,1]) and define ˜ f: [0,1]N→Ras ˜ f(x1,··· , xN) = f(xN). This function is clearly continuous on [0,1]N, in addition we have that graph(˜ f) = [0,1]N−1×graph(f) and by applying Lemma 2.7 with F= [0,1]N−1, it holds that dimH(graph(˜ f)) = N−1 + dimH(graph(f)). Now fix S∈(N, N + 1] and let Mbe the infinite-dimensional closed vector space of C([0,1]) given by Theorem 2.3 for s=S−N+ 1 ∈(1,2]. It is straightforward that the set f M:= {˜ f:f∈M}is an infinite dimensional closed vector space of C([0,1]N) all of whose nonzero functions have graph with Hausdorff dimension N−1 + s=S. A similar construction, based on Theorem 2.6, gives the maximal-dense lineability.  By the other hand, one may think that the same arguments as those used in Theorem 2.3 would imply that the set of continuous functions on [0,1] with prescribed Box dimension s∈(1,2] is spaceable. This is not true because, as it is shown in [13, Example 3.5], the Box dimension is not stable under numerable (disjoint) unions. In [5, Corollary 8] Baranski construct for every h, b ∈[1,2] with h≤ba function f∈ C([0,1]) such that dimH(graph(f)) = dimB(graph(f)) = hand dimB(graph(f)) = b. As a matter of fact, using this function, the stability of the Box dimension under finite unions and Lemma 2.5, the following result can also be proved. HAUSDORFF AND BOX DIMENSIONS VIA LINEABILITY 9 Corollary 2.9. Given h, b ∈[1,2] with h≤b, the set of functions f∈ C[0,1] with dimH(graph(f)) = dimB(graph(f)) = hand dimB(graph(f)) = b is dense-lineable in C[0,1]. In order to motivate the next question, we prove the next result. Proposition 2.10. Let g∈C1(R)be non-constant function such that Rg= {x∈R:g0(x) = 0}is finite. Then for any continuous function f: [0,1] →R dimH(graph[0,1](g◦f)) = dimH(graph[0,1](f)). In particular, dimH(graph[0,1](fn)) = dimH(graph[0,1](f)) for all n. Proof. First, we introduce the notation of convex hull. Namely, for a Borel set A⊂[0,1] let conv(A) the smallest closed interval such that A⊂conv(A). Observe that for any Borel set A⊂[0,1] and any bi-Lipschitz map g: conv(f(A)) →g(conv(f(A))) (2.1) dimHgraphA(g◦f) = dimHgraphA(f). Indeed, G: (x, y)→(x, g(y)) is also a bi-Lipschiz map on conv(A)× conv(f(A)), moreover graphA(g◦f) = G(graphA(f)). Equation (2.1) follows by the bi-Lipschitz invariance of the Hausdorff dimension. Let g∈C1(R) be non-constant function such that Rg={x∈R:g0(x) = 0}is finite. Then there exists Nsuch that for every n≥N,f([0,1] \ ∪y∈Rg(y−1/n, y + 1/n) is a finite union of disjoint closed intervals. Denote In 1,··· , In mthis collection of disjoint closed intervals. It is easy to see that g:In l→g(In l) is a bi-Lipschitz map. Moreover, [0,1] = {x:f(x)∈Rg}∪ ∞ [ n=N m [ l=1 f−1(In l). by (2.1), for every n≥Nand 1 ≤l≤m dimH(graphf−1(In l)(g◦f)) = dimH(graphf−1(In l)(f)). Hence, by the countable stability of the Hausdorff dimension dimH(graph[0,1](g◦f)) =dimHgraph{x:f(x)∈Rg}(g◦f)∪∞ [ n=N m [ l=1 graphf−1(In l)(g◦f) =maxdimHgraph{x:f(x)∈Rg}(g◦f), supn≥N,1≤l≤mdimHgraphf−1(In l)(g◦f) =maxdimHgraph{x:f(x)∈Rg}(f), supn≥N,1≤l≤mdimHgraphf−1(In l)(f) =dimHgraph{x:f(x)∈Rg}(f)∪∞ [ n=N m [ l=1 graphf−1(In l)(f) =dimH(graph[0,1](f)).