1 Introduction

Quasiarihmetic means were introduced in 1920-s/30-s by de Finetti [7], Knopp [12], Kolmogorov [13] and Nagumo [16]. For a continuous and strictly monotone function \(F :I \rightarrow \mathbb {R}\) (here and below I stands for an arbitrary subinterval of \(\mathbb {R}\) and \({{\,\mathrm{\mathcal{C}\mathcal{M}}\,}}(I)\) stands for a family of all continuous and strictly monotone functions on I) we define the quasiarithmetic mean \(\mathscr {A}^{[F]} :\bigcup _{n=1}^\infty I^n \rightarrow I\) by

$$\begin{aligned} \mathscr {A}^{[F]}(a):=F^{-1} \bigg ( \frac{F(a_1)+\cdots +F(a_n)}{n} \bigg ) \quad \text { where }n \in \mathbb {N}\text { and }a=(a_1,\ldots ,a_n) \in I^n. \end{aligned}$$

A function F is called a generating function or a generator of \(\mathscr {A}^{[F]}\). To avoid misunderstandings let introduce formally \(\mathbb {N}:=\{1,2,\ldots \}\) and \(\mathbb {N}_0:=\{0,1,2,\ldots \}=\mathbb {N}\cup \{0\}\).

It was Knopp [12] who noticed that for \(I=\mathbb {R}_+\), \(\pi _p(x):=x^p\) (\(p\ne 0\)) and \(\pi _0(x):=\ln x\), the quasiarithmetic mean \(\mathscr {A}^{[\pi _p]}\) coincides with the p-th power mean \(\mathscr {P}_p\).

Adapting the classical result \(\lim _{p \rightarrow +\infty } \mathscr {P}_p = \max \) we say that a sequence \((F_n)_{n=1}^\infty \) in \({{\,\mathrm{\mathcal{C}\mathcal{M}}\,}}(I)\) is QA-maximal provided

$$\begin{aligned} \lim _{n \rightarrow \infty }\mathscr {A}^{[F_n]} = \max \quad \text {pointwise.} \end{aligned}$$

There are few approaches to this property. First, applying some general results by Páles [27], we can establish the general equivalent condition for being QA-maximal (see also [19]). More precisely, a sequence \((F_n)_{n=1}^\infty \) of elements in \({{\,\mathrm{\mathcal{C}\mathcal{M}}\,}}(I)\) is a QA-maximal sequence if and only if

$$\begin{aligned} \lim _{n \rightarrow \infty }\frac{F_n(x)-F_n(y)}{F_n(z)-F_n(y)} =0 \quad \text { for all }x,\,y,\,z \in I\text { with }x<y<z. \end{aligned}$$

It is a particular case of an analogous result for deviation and quasideviation means – see the papers by Daróczy [2, 3], Daróczy–Losonczi [4], Daróczy–Páles [5, 6], and by Páles [20,21,22,23,24,25,26] for detailed study of these families.

It turns out that under the additional assumption that each generator is twice continuously differentiable with nowhere vanishing first derivative – from now on we denote family of all such generators by \(\mathcal {C}^{2\#}(I)\) – we can establish another equivalent conditions. More precisely, by [19], a sequence \((F_n)_{n=1}^\infty \) of elements in \(\mathcal {C}^{2\#}(I)\) such that \(\frac{F_n''}{F_n'}\) is uniformly lower bounded is QA-maximal if and only if

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _x^y \frac{F_n''(t)}{F_n'(t)}dt=+\infty \quad \text { for all }x,y \in I \text { with }x<y. \end{aligned}$$

Let us emphasize that the operator \(\frac{F''}{F'}\) plays a key role in a comparability of quasiarithmetic means. More precisely, by Jensen inequality, for all \(F,G \in \mathcal {C}^{2\#}(I)\) we have

$$\begin{aligned} \mathscr {A}^{[F]}\le \mathscr {A}^{[G]} \iff \frac{F''}{F'} \le \frac{G''}{G'}. \end{aligned}$$

In view of [28] if \(F_1,F_2,\ldots \in \mathcal {C}^{2\#}(I)\) such that \(\mathscr {A}^{[F_1]}\le \mathscr {A}^{[F_2]} \le \cdots \) then, by [17], the maximal property is connected with the set

$$\begin{aligned} \mathscr {I}_F:=\bigg \{x \in I : \lim _{n\rightarrow \infty } \frac{F_n''(x)}{F_n'(x)}=+\infty \bigg \}. \end{aligned}$$

Namely, it was proved that if \(\mathscr {I}_F=I\) then \((F_n)_{n=1}^\infty \) is QA-maximal. Conversely, for every QA-maximal sequence \((F_n)_{n=1}^\infty \) the set \(\mathscr {I}_F\) is a dense subset of I.

Observe that \(\mathscr {A}^{[F]}=\mathscr {A}^{[G]}\) if and only if \(\frac{F''}{F'} = \frac{G''}{G'}\), therefore we can associate the quasi-arithmetic mean with the value of this operator [17].

These results were strengthened in [19]. More precisely, there is proved that if the intersection of \(\mathscr {I}_F\) with an arbitrary open subset of I has a positive Lebesgue measure \(\lambda \), then \((F_n)_{n=1}^\infty \) is QA-maximal. This assumption is somehow the weakest possible, as for every \(X \subset I\) such that \(\lambda (X \cap J)=0\) for some open subinterval J of I there exists a sequence \((G_n)_{n=1}^\infty \) which is not QA-maximal, however \(X \subset \mathscr {I}_G\). On the other hand, it was proved that \(\mathscr {I}_Z\) could have the Hausdorff dimension zero for some QA-maximal sequence \((Z_n)_{n=1}^\infty \).

In what follows our aim is to study the relation between the property of being a max-family and the corresponding set \(\mathscr {I}\).

Let us emphasize that the same consideration remains valid for QA-minimal and \(\min \)-families (with a natural definition). As a matter of fact we can reapply all results below to the reflected means—for detailed study of reflected means we refer the reader to recent development by Chudziak–Páles–Pasteczka [1], Páles–Pasteczka [29] and Pasteczka [18].

1.1 Rephrasing of the problem

As all conditions above are expressed in terms of the operator \(F \mapsto F''/F'\) we are going to elaborate the properties of this operator. To this end, I stands for arbitrary interval in \(\mathbb {R}\) (unless explicitly stated otherwise) and \(\mathcal {C}(I)\) stands for a family of all continuous functions mapping an interval I into \(\mathbb {R}\). Moreover, for a sequence \(f=(f_1,f_2,\ldots )\) of continuous functions \(f_1,f_2,\ldots \in \mathcal {C}(I)\) with \(f_1\le f_2 \le \cdots \) define

$$\begin{aligned} I_f:=\big \{x \in I : \lim _{n\rightarrow \infty } f_n(x)=+\infty \big \}. \end{aligned}$$

Furthermore, let \(\Omega (I)\) be a family of all possible \(I_f\)-s. More precisely,

$$\begin{aligned} \Omega (I):=\big \{ I_f : f_1,f_2,\ldots \in \mathcal {C}(I) \text { and }f_1\le f_2 \le \cdots \big \}. \end{aligned}$$

Additionally, if

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _x^y f_n(t)dt=+\infty \quad \text { for all }x,y \in I \text { with }x<y, \end{aligned}$$
(1.1)

then the sequence \((f_n)_{n=1}^\infty \) is called max-family. Based on the previous section we obtain the following results.

Proposition 1.1

([19, Proposition 4.1]) Let \(X\subset I\) be an arbitrary set. If \(\lambda (X \cap J)=0\) for some open interval J, then there exists a sequence \((f_n)_{n=1}^\infty \) of functions \(f_1, f_2,\ldots \in \mathcal {C}(I)\) with \(f_1 \le f_2 \le \cdots \) which is not a max-family, although \(I_f \supset X\).

Proposition 1.2

( [19, Proposition 4.2]) Let \((f_n)_{n=1}^\infty \) be a sequence of functions \(f_1, f_2,\ldots \in \mathcal {C}(I)\) with \(f_1 \le f_2 \le \cdots \). If \(\lambda (I_f\cap J)>0\) for each open subinterval \(J\subset I\), then \((f_n)_{n=1}^\infty \) is a max-family.

Propositions above motivates us to define

$$\begin{aligned} \Omega _0(I):=\big \{ I_f\in \Omega (I) : (f_n)_{n=1}^\infty \text { is a max-family} \big \}. \end{aligned}$$

Obviously \(\Omega _0(I) \subseteq \Omega (I)\). The aim of this paper is to show several important facts concerning the set \(\Omega (I)\), \(\Omega _0(I)\), and their common relations.

2 General properties of \(\Omega (I)\) and \(\Omega _0(I)\)

First, let us present our initial result which shows that \(\Omega (I)\) consists of \(G_\delta \) sets only.

Lemma 2.1

Let \((f_n)_{n=1}^\infty \) be a sequence of functions \(f_1, f_2,\ldots \in \mathcal {C}(I)\) with \(f_1 \le f_2 \le \cdots \). Then \(I_f\) is a \(G_\delta \) set.

Proof

Set \(A_{n,M}:=\{x\in I : f_n(x)>M\}\) where \(M, n\in \mathbb {N}\). Then \(A_{n,M}\) are open and

$$\begin{aligned} x\in I_f\iff x\in \bigcap _{M\in \mathbb {N}}\bigcup _{N\in \mathbb {N}}\bigcap _{n\ge N}A_{n,M}. \end{aligned}$$

But the sequence \((f_n)_{n=1}^\infty \) is monotone, thus \(\bigcap _{n\ge N}A_{n,M}=A_{N,M}\) and \(I_f\) is \(G_\delta \). \(\square \)

Let us now prove that the set \(\Omega (I)\) is closed under finite union and closed under countable intersection (with additional assumption).

Lemma 2.2

Let \((f_n)_{n=1}^\infty \) and \((g_n)_{n=1}^\infty \) be two sequences of functions from \(\mathcal {C}(I)\) with \(f_1\le f_2\le \cdots \) and \(g_1\le g_2\le \ldots .\) Then \(I_{f+g}=I_f\cup I_g\).

Proof

Observe that as \(f_n\ge f_1\) and \(g_n\ge g_1\) we obtain \(I_f\subseteq I_{f+g}\) and \(I_g\subseteq I_{f+g}\). Therefore \(I_f \cup I_g \subseteq I_{f+g}\).

To prove the converse inclusion take \(x \in I_{f+g}\) arbitrarily. Then we have

$$\begin{aligned} +\infty =\lim _{n \rightarrow \infty }\big ( f_n(x)+g_n(x)\big )=\lim _{n \rightarrow \infty } f_n(x)+\lim _{n \rightarrow \infty }g_n(x), \end{aligned}$$

which shows that \(\lim _{n \rightarrow \infty } f_n(x)=+\infty \) or \(\lim _{n \rightarrow \infty }g_n(x)=+\infty \), i.e. \(x \in I_f \cup I_g\). \(\square \)

Corollary 2.3

Let \(I \subset \mathbb {R}\) be an arbitrary interval. Then

  1. (i)

    for all \(J, K \in \Omega (I)\) we have \(J \cup K \in \Omega (I)\);

  2. (ii)

    for all \(J \in \Omega (I)\) and \(K \in \Omega _0(I)\) we have \(J\cup K \in \Omega _0(I)\).

Observe that Lemma 2.2 is not true for infinitely many sequences of continuous functions. To see that take sequences \(\big (f^{(1)}_n\big )_{n=1}^\infty \), \(\big (f^{(2)}_n\big )_{n=1}^\infty \), ... and \(\big (g^{(1)}_n\big )_{n=1}^\infty \), \(\big (g^{(2)}_n\big )_{n=1}^\infty \), ... of functions on \(\mathbb {R}\) defined by

$$\begin{aligned} f_n^{(1)}&\equiv n,&\quad \text { for all }n \ge 1;\\ f_n^{(i)}&\equiv -1,&\quad \text { for all }i \ge 2 \text { and } n\ge i-1,\\ f_n^{(i)}&\equiv -2,&\quad \text { for all }i \ge 2 \text { and } n<i-1,\\ g_n^{(i)}&\equiv \frac{1}{i},&\quad \text { for all }i \ge 1 \text { and }n\ge i,\\ g_n^{(i)}&\equiv 0,&\quad \text { for all }i \ge 2 \text { and }n< i. \end{aligned}$$

Then \(\sum _if^{(i)}_n(x)=-\infty \) for each \(x\in \mathbb {R}\) and \(n\ge 1\) and thus we have \(I_{\sum _i f^{(i)}}=\emptyset \). On the other hand, in since \(I_{f^{(1)}}=\mathbb {R}\), we have \(\bigcup _iI_{f^{(i)}}=\mathbb {R}\).

For the second family one has \(\sum _ig^{(i)}_n\equiv \sum _{i=1}^n\frac{1}{i}=+\infty \), hence \(I_{\sum _i g^{(i)}}=\mathbb {R}\). However each \((g_n^{(i)})_{n=1}^\infty \) is uniformly bounded (for example by 1). Thus \(I_{g^{(i)}}=\emptyset \) for all \(i\in \mathbb {N}\) and so \(\bigcup _{i=1}^\infty I_{g^{(i)}}=\emptyset \). Above examples illustrate that both inclusions may be not valid in general.

Let us note as a curiosity the following remark that mimics Lemma 2.2.

Remark 2.4

Let \((f_n)_{n=1}^\infty \), \((g_n)_{n=1}^\infty \) be two sequences of functions from \(\mathcal {C}(I)\) with \(f_1 \le f_2 \le \cdots \) and \(g_1 \le g_2 \le \cdots \). Then \(I_{f\cdot g}= I_f\cup I_g\) provided that if \(x\in I_f\) (\(x\in I_g\)), then there is \(\eta \in (0,+\infty ]\) such that \(g_n(x)\rightarrow \eta \) (\(f_n(x)\rightarrow \eta \)).

Proof

Take an \(x \in I_{f\cdot g}\) arbitrarily. Then we have

$$\begin{aligned} +\infty =\lim _{n \rightarrow \infty } \big (f_n(x)\cdot g_n(x)\big ) \end{aligned}$$

which is possible when \(\lim _{n \rightarrow \infty } f_n(x)\) and \(\lim _{n \rightarrow \infty }g_n(x)\) are positive and at least one of them is \(+\infty \). Hence \(x \in I_f \cup I_g\).

To show the converse inclusion assume that \(x\in I_f\) and \(\lim \limits _{n\rightarrow \infty }g_n(x)=\eta >0\). Then,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\big (f_n(x)\cdot g_n(x)\big )= \lim \limits _{n\rightarrow \infty }f_n(x)\cdot \lim \limits _{n\rightarrow \infty }g_n(x)=+\infty \cdot \eta =+\infty \end{aligned}$$

and \(x\in I_{f\cdot g}\). Thus \(I_f\subseteq I_{f\cdot g}\) and similarly, \(I_g\subseteq I_{f\cdot g}\). \(\square \)

Lemma 2.5

Let I be a closed interval and \((D_n)_{n=0}^\infty \) be a sequence of closed subsets of I such that \(D_0=I\) and \(D_{n+1} \subset {{\,\textrm{int}\,}}D_n\) for all \(n \in \mathbb {N}_0\). Then \(\bigcap _{n=0}^\infty D_n \in \Omega (I)\).

Proof

Indeed, in view of Tietze (Urysohn-Brouwer) theorem, for every \(n \in \mathbb {N}_0\) there exists a continuous function \(\delta _n :I \rightarrow [0,1]\) such that

$$\begin{aligned} \delta _n(x)= {\left\{ \begin{array}{ll} 0 &{} \text { for }x \in I \setminus D_n; \\ 1 &{} \text { for }x \in D_{n+1}. \end{array}\right. } \end{aligned}$$

Define \(d_n:=\sum _{i=0}^n \delta _n\). Then for every \(x \in \bigcap _{k=0}^\infty D_k=:D_\infty \) we have \(d_n(x)=n\). In particular \(I_d \supseteq D_\infty \). Now fix \(N \in \mathbb {N}\). Then as \(D_n \subseteq D_{N}\) for \(n \ge N\), we have \(\delta _n(x)=0\) for \(x\in I\setminus D_{N}\). Thus

$$\begin{aligned} d_n(x)<N \quad \text { for all }n \ge N \text { and }x \in I \setminus D_{N}. \end{aligned}$$

This proves that \(I_d \subseteq D_{N}\). As N was an arbitrary natural number we get \(I_d \subseteq D_\infty \), which completes the equality \(I_d=\bigcap _{k=0}^\infty D_n\). Therefore \(\bigcap _{k=0}^\infty D_n \in \Omega (I)\). \(\square \)

Lemma 2.6

([19], Proposition 4.3) For every interval I there exists a max-family \((z_n)_{n=1}^\infty \), \(z_n:I\rightarrow \mathbb {R}_+\), such that \(\dim _H I_z=0\).

In particular, \(\Omega _0(I)\) contains a set of zero Hausdorff dimension.

In the next lemma we show how some simple transformation of the interval I affect respective sets \(\Omega (\cdot )\) or \(\Omega _0(\cdot )\). Our aim is to reduce all cases to the unit interval.

Lemma 2.7

Let \(I \subset \mathbb {R}\) be an interval. Then

  1. (i)

    \(\Omega (aI+b)=\{aQ+b : Q \in \Omega (I)\}\) for all \(a, b \in \mathbb {R}\), \(a \ne 0\);

  2. (ii)

    \(\Omega (J)=\big \{Q \cap J : Q \in \Omega (I)\big \}\) for every subinterval J of I;

  3. (iii)

    \(\Omega (I \cup \{x_0\})=\Omega (I) \cup \big \{ Q\cup \{x_0\} : Q \in \Omega (I)\big \}\) for \(x_0\) being an endpoint of I.

All statements above remains valid with \(\Omega \) replaced by \(\Omega _0\).

Proof

The first statement is trivial as we may apply the affine transformation for each function.

Now let J be a subinterval of I. Then for every element \(W \in \{Q\cap J : Q\in \Omega (I)\}\) there exists a sequence \((f_n)_{n=1}^\infty \) of functions \(f_1,f_2,\ldots \in \mathcal {C}(I)\) such that \(f_1\le f_2 \le \cdots \) and \(W=I_f\cap J\).

Now let \(g_i=f_i|_J\) for all \(i \in \mathbb {N}\). Then obviously \(I_g=I_f \cap J\), and \((g_i)_{i=1}^{\infty }\) is a nondecrasing sequence of continuous functions. Consequently \(\Omega (J) \ni I_g=I_f\cap J=W\), which yields \(\{Q\cap J : Q\in \Omega (I)\} \subseteq \Omega (J)\).

To show the converse inclusion observe that for every \(W \in \Omega (J)\) there exists a sequence \(g_1,g_2,\ldots \in \mathcal {C}(J)\) such that \(0\equiv g_1\le g_2 \le \cdots \) and \(\{x \in J : \lim _{n \rightarrow \infty } g_n(x)=+\infty \}=W\).

For \(k \ge 2\) define \(\delta _k:J \rightarrow [0,+\infty )\) by \(\delta _k := g_{k}-g_{k-1}\) and set \(\delta _1\equiv 0\). Then each \(\delta _k\) admit a continuous prolongation \(\hat{\delta }_k:I \rightarrow [0,+\infty )\) (\(k \in \mathbb {N}\)) such that

$$\begin{aligned} \hat{\delta }_k(x)=1 \quad \text {for all }x \in I \text { with }{{\,\textrm{dist}\,}}(x, J)>\tfrac{1}{k}. \end{aligned}$$

We can use the latter extension in order to generate a sequence \((\hat{g}_n)_{n=1}^\infty \) of continuous functions on I by \(\hat{g}_n:=\hat{\delta }_1+\cdots +\hat{\delta }_{n-1}\). Then, with the natural extension of the notion of sum, we have \(0 = \hat{g}_1\le \hat{g}_2 \le \cdots \) and \(\hat{g}_ k|_J=g_k\) for all admissible k-s. Thus \(I_{\hat{g}} \cap J=W\), completing the proof of (ii). Case (ii) with \(\Omega \) replaced by \(\Omega _0\) is analogous.

To show (iii) assume that \(x_0\) is a boundary point of I and \(x_0 \notin I\). Now the \((\subseteq )\) inclusion of (iii) is a straightforward implication of the first part. Indeed, for every element \(X \in \Omega (I \cup \{x_0\})\), by (ii), we know that \(X\cap I \in \Omega (I)\). Thus either \(x_0 \notin X\) and \(X=X\cap I \in \Omega (I)\) or \(x_0 \in X\) and \(X\cap I=X \setminus \{x_0\}\). In the second case \(X=(X\cap I) \cup \{x_0\}\) which, since \(X \cap I \in \Omega (I)\) yields \(X\in \big \{ Q\cup \{x_0\} : Q \in \Omega (I)\big \}\). Upon binding these case we obtain \(X \in \Omega (I) \cup \big \{ Q\cup \{x_0\} : Q \in \Omega (I)\big \}\). Thus

$$\begin{aligned} \Omega (I \cup \{x_0\}) \subseteq \Omega (I) \cup \big \{ Q\cup \{x_0\} : Q \in \Omega (I)\big \}. \end{aligned}$$

Furthermore \(\Omega (I) \subset \Omega (I \cup \{0\})\). Thus the only assertion left to show is

$$\begin{aligned} Q \in \Omega (I) \ \text { implies } \ Q \cup \{x_0\} \in \Omega (I \cup \{x_0\}). \end{aligned}$$
(2.1)

To this end, take a sequence of functions \(f_1,f_2,\ldots \in \mathcal {C}(I)\) such that \(f_1\le f_2 \le \cdots \) and \(I_f=Q\), and \(h_k\in \mathcal {C}(I)\) by

$$\begin{aligned} h_k(x)=\max \{0,k-|k^4(x-x_0)|\} \quad n \in \mathbb {N}. \end{aligned}$$

Let \(g_n(x)=\sum _{k=1}^nh_k(x)\). Then \(I_g=\{x_0\}\) and thus \(I_{f+g}=Q\cup \{x_0\}\), i.e. \(Q\cup \{x_0\} \in \Omega (I\cup \{x_0\})\) which completes the proof of (2.1).

Moreover we have

$$\begin{aligned} \Vert h_k\Vert _1 \le \int _{-\infty }^{+\infty }h_k(x)\,dx = \int _{x_0-\tfrac{1}{k^3}}^{x_0+\tfrac{1}{k^3}}h_k(x)\,dx=\frac{2}{k^2} \quad \text {for all }k \in \mathbb {N}, \end{aligned}$$

which yields

$$\begin{aligned} \Vert g_n\Vert _1 =\Big \Vert \sum _{k=1}^n h_k\Big \Vert _1 \le \sum _{k=1}^n\Vert h_k\Vert _1\le \sum _{k=1}^n \frac{2}{k^2}< \frac{\pi ^2}{3} \quad \text {for all }n\in \mathbb {N}, \end{aligned}$$

and shows that (2.1) is valid with \(\Omega \) replaced by \(\Omega _0\). \(\square \)

3 Max-families with noninteger Hausdorff dimension

In the theory of fractals two natural questions are present: Is there a set which Hausdorff dimension equals the given number? And if the answer is positive: What additional features this set can have? The answers to the first question was given for instance by [10, 30, 31], while the answer to the second question depends on the feature (see [32] for connections with ergodicity and continued fractions, [9] for properties of distance sets and [14] for examples of subrings of \(\mathbb {R}\)).

In this section we present two construction of max-families with an arbitrary Hausdorff dimension \(\theta \in (0,1)\). In the first (Cantor-type) approach we show that \(\Omega _0(I)\) contains a set which can be factorized to a nowhere dense set of Hausdorff dimension \(\theta \) and a dense set of Hausdorff dimension zero. In the second (Jarník-type) approach we construct a set in \(I_f \in \Omega _0(I)\) such that \(\dim _H(I_f\cap U)=\theta \) for an arbitrary open interval \(U \subset I\). We provide two constructions; the Cantor-like sets can be described directly, while the second one relies on number-theoretical approach and thus does not give any insight on how does the provided set actually look like.

3.1 Cantor-type construction

We recall some basic notation and definitions from the fractal theory. We call a function \(f:X\rightarrow X\) a contraction if it is Lipschitz with constant \(c_f\in (0,1)\). We call a finite set \(\mathcal {F}=\{f_1, \ldots , f_k\}\) of contractions defined on a compact metric space X an iterated function system, or IFS. We say that the IFS \(\mathcal {F}\) satisfies the open set condition (abbreviated OSC) if there exists an open and bounded set \(V\ne \emptyset \) with \(F(V)\subset V\), where \(F(V):=f_1(V)\cup \ldots \cup f_k(V)\) and \(f_i(V)\cap f_j(V)=\emptyset \) for all distinct \(i,j \in \{1,\ldots k\}\). A contraction \(f :X \rightarrow X\) is called a similarity if \(\Vert f(x)-f(y)\Vert =c_f\Vert x-y\Vert \) for all \(x,y \in X\).

Proposition 3.1

(Moran [15]) Fix \(k,n\in \mathbb {N}\). Suppose that \(\mathcal {F}=\{f_1\ldots ,f_k\}\) satisfies the open set condition and each \(f_i\in \mathcal {F}\), \(i=1,\ldots ,k\), is a similarity on \(\mathbb {R}^n\) with contraction constant \(c_i\). If A is the compact set such that \(F(A)=A\), then \(\dim _HA=s\), where s is the unique solution of the equation

$$\begin{aligned} \sum \limits _{i=1}^n c_i^s=1. \end{aligned}$$

The following Lemma is a well-known construction of Cantor-type set with given Hausdorff dimension. We give a proof in full detail.

Lemma 3.2

For every compact interval I and every \(\theta \in (0,1)\) there exists a sequence \((D_n)_{n=0}^\infty \) of closed sets such that

  1. (i)

    \(D_0=I\),

  2. (ii)

    \(D_{n+1} \subset {{\,\textrm{int}\,}}D_n\) for all \(n \in \mathbb {N}_0\),

  3. (iii)

    \(D_\infty :=\bigcap _{n=0}^\infty D_n\) is an invariant set of some IFS,

  4. (iv)

    \(\dim _H D_{\infty }=\theta \).

Proof

One can assume without loss of generality that \(I=[0,1]\). Let \(m:=(\tfrac{1}{2})^{1/\theta } \in (0,\frac{1}{2})\). Take \(\varepsilon \in (0,\tfrac{1}{2m}-1)\) and define an IFS \(\mathcal {F}=\{F_L, F_R\}\) on I by

$$\begin{aligned} F_L(x):=m \cdot (x+\varepsilon ) \quad F_R(x):=1-F_L(x). \end{aligned}$$

Then \(F_L\) and \(F_R\) are similarities with Lipschitz constants m, \(F_L(I)=[m\varepsilon ,m(1+\varepsilon )]\) and \(F_R(I)=[1-m(1+\varepsilon ),1-m\varepsilon ]\).

But

$$\begin{aligned} \varepsilon<\tfrac{1}{2m}-1 \iff 2m(1+\varepsilon )<1 \iff m(1+\varepsilon )<1-m(1+\varepsilon ), \end{aligned}$$

which yields \(F_L(I) \cap F_R(I)=\emptyset \) and thus \(\mathcal {F}\) satisfies the OSC with \(U={{\,\textrm{int}\,}}I\). Now define a sequence \((D_n)_{n=0}^\infty \) by

$$\begin{aligned} D_0:= & {} I , \\ D_{n+1}:= & {} F_L(D_n)\cup F_R(D_n) \text { for } n \ge 0. \end{aligned}$$
Fig. 1
figure 1

Construction of \(D_n\)’s

Full size image

Obviously each \(D_n\) is a closed subset of I. Now we prove that the sequence \((D_n)_{n=0}^\infty \) satisfies all conditions (i)–(iv). Condition (i) is obvious. To show the second property observe that

$$\begin{aligned} D_1 =\big [m\varepsilon ,m(1+\varepsilon )\big ] \cup \big [1-m(1+\varepsilon ),1-m\varepsilon \big ] \subset (0,1)={{\,\textrm{int}\,}}D_0. \end{aligned}$$

Moreover if \(D_{n+1} \subset {{\,\textrm{int}\,}}D_n\) for some \(n \in \mathbb {N}\), then as both \(F_L\) and \(F_R\) are homeomorphisms we get

$$\begin{aligned} F_L(D_{n+1}) \subset {{\,\textrm{int}\,}}F_L(D_n) \quad \text{ and } \quad F_R(D_{n+1}) \subset {{\,\textrm{int}\,}}F_R(D_n), \end{aligned}$$

and thus \(D_{n+2} \subset {{\,\textrm{int}\,}}D_{n+1}\). By simple induction we obtain property (ii).

Denote \(D_\infty =\bigcap _{n=0}^\infty D_n\). Condition (iii) follows from (ii) and the general theory of fractal sets (see for instance Theorem 9.1 in [8]).

To check the last condition denote \(s:=\dim _H D_\infty \). Then, by Proposition 3.1 we have \(2m^s=1\). Thus \(s=\frac{\ln (1/2)}{\ln m}=\theta \), which is (iv). \(\square \)

Remark 3.3

In Lemma 3.2 the set \(D_\infty \) has Lebesgue measure zero. It turns out that the entire construction presented there is equivalent to the construction of the uniform Cantor set on the interval \([m\varepsilon /(1-m),1-m\varepsilon /(1-m)]\) with the middle part of length \(\frac{(1-2m)(1-m-2m\varepsilon )}{1-m}\) removed (that is both constructions lead to the same set).

Lemma 3.4

For every \(\theta \in [0,1]\) there exists a sequence \((d_n )_{n=1}^\infty \) of functions from \(\mathcal {C}(I)\) such that \(0\le d_1\le d_2 \le \cdots \), the set \(I_d\) is nowhere dense, and \(\dim _HI_d=\theta \).

Proof

For \(\theta =0\) the statement is an easy implication of Lemma 2.6. For \(\theta \in (0,1)\) set a sequence \((D_n)_{n=0}^\infty \) like in Lemma 3.2. By Lemma 2.5 we get that \(I_d=\bigcap _{n=0}^\infty D_n \in \Omega (I)\) which is easily equivalent to our statement.

For \(\theta =1\) we can apply the construction in Lemma 3.2 to Smith–Volterra–Cantor set, that is we mimic the construction of Cantor-type set in Lemma 3.2 with appropriate changes to obtain Smith–Volterra–Cantor set (cf. Remark 3.3). Such a set is nowhere dense and has Hausdorff dimension 1. \(\square \)

Theorem 3.5

For every \(\theta \in [0,1]\), there exists a max-family \((f_n)_{n=1}^\infty \) of functions from \(\mathcal {C}(I)\) such that \(\dim _HI_f=\theta \) and \(I_f\) can be decomposed to a nowhere dense set and a set of Hausdorff dimension zero.

Proof

Take \(\theta \in [0,1]\) arbitrarily. By Lemma 2.6 one can take a max-family \((z_n)_{n=1}^\infty \), where \(z_n:I\rightarrow \mathbb {R}_+\) for \(n\ge 1\), such that \(\dim _HI_z=0\). Furthermore, by Lemma 3.4 we can take a sequence \((d_n)_{n=1}^\infty \) of functions from \(\mathcal {C}(I)\) such that \(I_d\) is nowhere dense, \(\dim _HI_d=\theta \), and

$$\begin{aligned} 0\le d_1\le d_2\le \cdots . \end{aligned}$$
(3.1)

Let \(f_n:=d_n+z_n\) for all \(n \in \mathbb {N}\). Then, by Lemma 2.2, we have \(\dim _HI_f=\theta \) and \(I_f=I_d\cup I_z\) admit a decomposition mentioned in the statement.

Therefore it is sufficient to show that \((f_n)_{n=1}^\infty \) is a max-family. As \((z_n)_{n=1}^\infty \) is a max-family we have \(0\le z_1\le z_2 \le \cdots \). Binding this property with (3.1) and the definition of the sequence \((f_n)_{n=1}^\infty \) we easily obtain

$$\begin{aligned} 0 \le f_1 \le f_2 \le \cdots . \end{aligned}$$

Thus the only remaining part to be proved is that (1.1) holds. However, as \(d_n \ge 0\) for all \(n \in \mathbb {N}\) and \((z_n)_{n=1}^\infty \) is a max-family, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _x^y f_n(t)dt=\lim _{n\rightarrow \infty } \int _x^y (d_n+z_n)(t)dt \ge \lim _{n\rightarrow \infty } \int _x^y z_n(t)dt=+\infty \\ \text { for all }x,y \in I\text { with }x<y, \end{aligned}$$

which completes the proof. \(\square \)

Remark 3.6

The set \(I_f\) in Theorem 3.5 cannot have positive measure (compare with Proposition 1.1 and Proposition 1.2). It turns out that the set \(D_\infty \) obtained in Lemma 3.2 has zero Lebesgue measure. In fact all Borel sets D with \(\dim _HD\in (0,1)\) have zero measure. This is because the Hausdorff measure \(H^d\) is up to a constant equivalent to the Lebesgue one, i.e. for integer d, \(H^d(D)=c(d)\lambda ^d(D)\) where c(d) is a known constant. Hence, if \(\lambda ^1(D)>0\), then \(H^1(D)>0\) and thus \(\dim _HD=1\). Note that the condition \(\dim _HD=1\) can imply \(\lambda ^1(D)=0\).

3.2 Jarník-type construction

In what follows we show that for every \(\theta \in [0,1]\) there exists a max-family \((f_n)_{n=1}^\infty \) of functions from \(\mathcal {C}(I)\) such that \(\dim _H(I_f \cap J)=\theta \) for every open subinterval \(J \subset I\). For \(\theta =0\) this statement is true due to [19, Proposition 4.3]. For \(\theta =1\) it is trivial since one can construct a family \((f_n)_{n=1}^\infty \) for which \(I_f=I\). For the remaining values of \(\theta \) the important theorem concerning neighbourhoods of rational numbers will be used.

Proposition 3.7

(Jarník [11]) Suppose \(\alpha >2\). Let \(Q_\alpha \) be the set of real numbers \(x\in [0,1]\) for which the inequality

$$\begin{aligned} \Vert qx\Vert \le q^{1-\alpha } \end{aligned}$$

is satisfied by infinitely many positive integers q, where

$$\begin{aligned} \Vert y\Vert :=\min \limits _{z\in \mathbb {Z}}|y-z|. \end{aligned}$$

Then \(\dim _H Q_\alpha =2/\alpha \). Moreover, \(Q_\alpha \) is dense in [0, 1] and \(\dim _H(Q_\alpha \cap J)=2/\alpha \) for every subinterval \(J\subset [0,1]\).

Theorem 3.8

For every compact interval I and every \(\theta \in [0,1]\) there exists a max-family \((f_n)_{n=1}^\infty \) of functions from \(\mathcal {C}(I)\) such that \(\dim _H(I_f\cap U)=\theta \) for every open subinterval U of I.

Proof

The case \(\theta \in \{0,1\}\) holds (see the beginning of this section). From now on let us assume that \(\theta \in (0,1)\). Set \(I=[0,1]\) and \(\alpha _0:=\tfrac{2}{\theta }\). For \(q \in \mathbb {N}\) and \(\alpha >2\) define

$$\begin{aligned} Y_{q,\alpha }&:=\big \{x \in (0,1) : \Vert qx\Vert \le q^{1-\alpha }\big \},\\ Z_{q,\alpha }&:=\big \{x \in (0,1) : \Vert qx\Vert \le \tfrac{q+1}{q}q^{1-\alpha }\big \}. \end{aligned}$$

Then, as \(\Vert \cdot \Vert \) is continuous, we have \({{\,\textrm{cl}\,}}Y_{q,\alpha } \subset {{\,\textrm{int}\,}}Z_{q,\alpha }\) for all \(q \in \mathbb {N}\) and \(\alpha >\alpha _0\). Now let \(\delta _q :I \rightarrow [0,1]\) be defined by

$$\begin{aligned} \delta _q(x)= {\left\{ \begin{array}{ll} 0 &{} \text { for }x \in I \setminus Z_{q,\alpha _0}; \\ 1 &{} \text { for }x \in Y_{q,\alpha _0}. \end{array}\right. } \end{aligned}$$

Define \(r_n :I \rightarrow \mathbb {R}_+\) by \(r_n:=\delta _1+\dots +\delta _n\). Then we obtain

$$\begin{aligned} I_r\supseteq \{x \in I : x\in Y_{q,\alpha _0} \text { for infinitely many }q\in \mathbb {N}\}=Q_{\alpha _0}. \end{aligned}$$

On the other hand for all \(\alpha <\alpha _0\) there exists a number \(q_0 \in \mathbb {N}\) such that

$$\begin{aligned} \tfrac{q+1}{q} q^{1-\alpha _0}\le q^{1-\alpha } \quad \text { for all }q\ge q_0. \end{aligned}$$

Thus, for all \(\alpha \in (2,\alpha _0)\) we have

$$\begin{aligned} I_r\subseteq & {} \{x \in I : x\in Z_{q,\alpha _0} \text { for infinitely many }q\in \mathbb {N}\}\\= & {} \{x \in I : x\in Z_{q,\alpha _0} \text { for infinitely many }q\in \mathbb {N}\text { with }q \ge q_0\}\\\subseteq & {} \{x \in I : x\in Y_{q,\alpha } \text { for infinitely many }q\in \mathbb {N}\text { with }q \ge q_0\}\\= & {} \{x \in I : x\in Y_{q,\alpha } \text { for infinitely many }q\in \mathbb {N}\}=Q_\alpha . \end{aligned}$$

Finally we have

$$\begin{aligned} Q_{\alpha _0} \subseteq I_r \subseteq \bigcap _{\alpha \in (2,\alpha _0)} Q_\alpha . \end{aligned}$$

Therefore by Proposition 3.7 we obtain that for every open interval \(U \subset I\) we have

$$\begin{aligned} \tfrac{2}{\alpha _0}=\dim _H (Q_{\alpha _0} \cap U)&\le \dim _H(I_r \cap U) \le \inf _{\alpha \in (2,\alpha _0)} \dim _H( Q_\alpha \cap U) =\inf _{\alpha \in (2,\alpha _0)}\tfrac{2}{\alpha }=\tfrac{2}{\alpha _0}. \end{aligned}$$

Consequently, \(\dim _H(I_r \cap U)=\tfrac{2}{\alpha _0}=\theta \) for every open subinterval \(U \subset I\).

Now let \((z_n)_{n=1}^\infty \) be a sequence from Lemma 2.6 and let \(f_n:=z_n+r_n\) for \(n \in \mathbb {N}\). Then, as \(r_n \ge 0\), we obtain that \((f_n)_{n=1}^\infty \) is a max-family. Furthermore by Lemma 2.2 we get that \(I_f \supset I_r\) and \(I_f \setminus I_r\) is of Hausdorff dimension zero which completes the proof. \(\square \)

3.3 Final conclusions and remarks

At the very end let us put the reader’s attention to few important problems. First, we cannot exclude that \(\Omega (I)\) is a family of all \(G_\delta \) subsets of I and/or \(\Omega _0(I)\) contains all dense \(G_\delta \) subsets of I. In particular, it is interesting to find a full characterization of all elements of sets \(\Omega (I)\) and \(\Omega _0(I)\) (our results show that they can be complicated from the measure-theoretical point of view). Second, this problem has a natural multidimensional generalization where the domain of the integral in (1.1) is taken over all open subsets of a given domain. Finally, it is not known if the assumption \(f_1\le f_2 \le \cdots \) in the definition of \(\Omega (I)\) can be relaxed.