Fekete's lemma for componentwise subadditive functions of two or more real variables

TL;DR

This paper extends Fekete's lemma to functions of multiple variables that are subadditive in each variable separately, demonstrating boundedness on bounded sets and generalizing classical results.

Contribution

It introduces a multivariable analogue of Fekete's lemma for componentwise subadditive functions, expanding the classical one-variable case and previous partial results.

Findings

Proves an analogue of Fekete's lemma for multivariable functions

Shows such functions are bounded on bounded subsets

Extends classical subadditivity results to multiple variables

Abstract

We prove an analogue of Fekete's subadditivity lemma for functions of several real variables which are subadditive in each variable taken singularly. This extends both the classical case for subadditive functions of one real variable, and a result in a previous paper by the author. While doing so, we prove that the functions with the property mentioned above are bounded in every closed and bounded subset of their domain. The arguments follows those of Chapter 6 in E. Hille's 1948 textbook.

Equations

$$f(x\cdot y)⩽f(x)+f(y)$$

$$\frac{f(x_1,x_2)}{x_1\cdot x_2}<\underset{x_1,x_2>0}{\inf }\frac{f(x_1,x_2)}{x_1\cdot x_2}+\delta .$$

$$\underset{x_1\to +\infty }{\lim }\underset{x_2\to +\infty }{\lim }\frac{f(x_1,x_2)}{x_1\cdot x_2}=\underset{x_2\to +\infty }{\lim }\underset{x_1\to +\infty }{\lim }\frac{f(x_1,x_2)}{x_1\cdot x_2}=\underset{x_1,x_2>0}{\inf }\frac{f(x_1,x_2)}{x_1\cdot x_2}.$$

$$\underset{x\to \mathcal{U}}{\mathrm{liminf}}f(x)=\underset{x\in U}{\sup }\underset{y\succcurlyeq x}{\inf }f(y)$$

$$\underset{x\to \mathcal{U}}{\mathrm{limsup}}f(x)=\underset{x\in U}{\inf }\underset{y\succcurlyeq x}{\sup }f(y),$$

$$\underset{x\in U}{\inf }f(x)⩽\underset{x\to \mathcal{U}}{\mathrm{liminf}}f(x)⩽\underset{x\to \mathcal{U}}{\mathrm{limsup}}f(x)⩽\underset{x\in U}{\sup }f(x).$$

$$\underset{x\to \mathcal{U}}{\mathrm{liminf}}f(x)⩽\underset{y\to \mathcal{V}}{\mathrm{liminf}}g(y)⩽\underset{y\to \mathcal{V}}{\mathrm{limsup}}g(y)⩽\underset{x\to \mathcal{U}}{\mathrm{limsup}}f(x).$$

$$x{⩽}_{\Pi }y⟺x_i⩽y_i\text{for every }i\in I.$$

$$f(n_1,\dots ,n_d)={\log }_a{\mathcal{A}}_X(n_1,\dots ,n_d)\text{for every}{n}_1,\dots ,n_d\in {\mathbb{Z}}_{+}$$

$$f(x_1{y}_1,x_2{y}_2)⩽f(x_1,x_2)+f(y_1,y_2),$$

$$f(x_1{y}_1,x_2{y}_2)⩽f(x_1,x_2)+f(x_1,y_2)+f(y_1,x_2)+f(y_1,y_2).$$

$$h(x_i)=\underset{x_{j_1}\to +\infty }{\lim }\dots \underset{x_{j_k}\to +\infty }{\lim }\frac{f(x_1,\dots ,x_d)}{{\prod }_{i=1}^k{x}_{j_i}}$$

$$\frac{f(\dots ,x_i+x_i^{\prime },\dots )}{x_j}⩽\frac{f(\dots ,x_i,\dots )}{x_j}+\frac{f(\dots ,x_1^{\prime },\dots )}{x_j},$$

$$\underset{(x_1,\dots ,x_d)\to \mathcal{U}}{\lim }\frac{f(x_1,\dots ,x_d)}{x_1\cdots x_d}=\underset{x_1,\dots ,x_d\in {\mathbb{Z}}_{+}}{\inf }\frac{f(x_1,\dots ,x_d)}{x_1\cdots x_d}.$$

$$h(X)=\underset{(x_1,\dots ,x_d)\to \mathcal{U}}{\lim }\frac{{\log }_a{\mathcal{A}}_X(x_1,\dots ,x_d)}{x_1\cdots x_d}$$

$$\frac{f(x,y)}{x\cdot y}$$

$$V_{t_1,\dots ,t_d,k}=\{(x_1,\dots ,x_d)\in {\mathbb{R}}_{+}^d\mid 0<x_i<t_i\forall i\in \left[1:d\right],f(x_1,\dots ,x_d)⩾k\}.$$

$$\mu \left(V_{t_1,\dots ,t_d,\frac{f(t_1,\dots ,t_d)}{2^d}}\right)⩾\frac{t_1\cdots t_d}{2^d}.$$

$$(x_1,\dots ,x_d)\mapsto \left(y_1^{(w_1)},\dots ,y_d^{(w_d)}\right)$$

$$V_w=\left\{\left(y_1^{(w_1)},\dots ,y_d^{(w_d)}\right)\mid (x_1,\dots ,x_d)\in V\right\}$$

$$f(t_1,\dots ,t_d)⩽\sum _{w\in \{0,1{\}}^d}f\left(y_1^{(w_1)},\dots ,y_d^{(w_d)}\right).$$

$$f(t_1,t_2)$$

$$t_1\cdots t_d⩽\sum _{w\in \{0,1{\}}^d}\mu (V_w)=2^d\mu (V).$$

$$W_{x_{1,n},\dots ,x_{d,n}}\subseteq V_{b,\dots ,b,n},$$

$$\mu \left(W_{x_{1,n},\dots ,x_{d,n}}\right)⩾\frac{x_{1,n}\cdots x_{d,n}}{2^d}⩾{\left(\frac{a}{2}\right)}^d.$$

$$M=\sup \{f(z_1,\dots ,z_d)\mid (z_1,\dots ,z_d)\in J\},$$

$$f(y_1+x_{1,n},\dots ,y_d+x_{d,n})⩽(2^d-1)M-n,$$

$$f(y_1+x_{1,n},y_2+x_{2,n})$$

$$K=\prod _{i=1}^d[x_i+2,x_i+3]\subseteq J,$$

$${\mathbb{R}}^d=\left(\bigcup _{w\in \{0,1{\}}^d}{\mathbb{R}}_w\right)\cup \left(\bigcup _{i=1}^n{V}_i\right).$$

Full text

Mathematics Subject Classification 2010: 39B62, 26B35, 00A05.

Key words: subadditivity, multivariate analysis, simultaneous limit.

1 Introduction

A real-valued function $f$ defined on a semigroup $(S,\cdot)$ is subadditive if

[TABLE]

for every $x,y\in{S}$. Examples of subadditive functions include the absolute value of a complex number; the ceiling of a real number (smallest integer not smaller than it); the cardinality of a a finite subset of a given set; and the length of a word over an alphabet. Subadditive functions have applications in many fields including information theory [9], economics, and combinatorics. In real analysis, subadditive functions are much more interesting than additive ones: for example, every additive, Lebesgue measurable function of one real variable is linear (cf. [12]) but the characteristic function of the set of irrational numbers is subadditive, Lebesgue measurable, not linear, and everywhere discontinuous.

A classical result in mathematical analysis, Fekete’s lemma [3] states that, if $f$ is a real-valued subadditive function of one positive integer or positivereal variable, then $f(x)/x$ converges, for $x\to+\infty$, to its greatest lower bound. This simple fact has a huge number of applications in many fields, including symbolic dynamics (cf. [9, Chapter 4]) and the theory of neural networks (see [5]). Reusing a metaphor from [1], Fekete’s lemma says that for a sequence of independent observations, the average information per observation converges to its greatest lower bound.

Given the importance and ubiquity of Fekete’s lemma, we wonder if similar results may hold for functions of many variables. Oddly, the mathematical literature seems to contain generalizations where, in almost all cases, the function in the limit actually depends again on a single variable, which is sometimes a real number, sometimes a finite set; many of these are closer to a corollary than to an extension. Of course there are practical reasons for this: for one, division by a vector with many components is undefined. But maybe we could look for a different type of limit, or even for a different flavor of subadditivity.

At the aim of understanding which of the above could be feasible, we note that, if $S=S_{1}\times S_{2}$ is a product semigroup, we can also consider the case of a function which is subadditive in each variable, however given the other. That is, instead of requiring $f(x_{1}y_{1},x_{2}y_{2})\leqslant f(x_{1},x_{2})+f(y_{1},y_{2})$ for every $x_{1}$, $x_{2}$, $y_{1}$, and $y_{2}$, we could demand that:

$f(x_{1}y_{1},x_{2})\leqslant f(x_{1},x_{2})+f(y_{1},x_{2})$ for every $x_{1}$, $x_{2}$, and $y_{1}$; and 2. 2.

$f(x_{1},x_{2}y_{2})\leqslant f(x_{1},x_{2})+f(x_{1},y_{2})$ for every $x_{1}$, $x_{2}$, and $y_{2}$.

The two requirements above, even together, do not imply subadditivity as a function defined on the product semigroup, nor does the latter imply the former: see Example 3.4. Oddly again, this multivariate “componentwise subadditivity” seems not to have been addressed very often in the literature.

In this paper, we state and prove an extension of Fekete’s lemma to componentwise subadditive functions of $d\geqslant 2$ real variables. We state a special case as an example, leaving the full statement to Section 5.

Proposition 1.1**.**

Let $f$ be a function of two positive real variables which is subadditive in each of them, however given the other. For every $\delta>0$ there exists $R>0$ such that, if both $x_{1}>R$ and $x_{2}>R$, then:

[TABLE]

In addition,

[TABLE]

That is: if the componentwise subadditive function $f(x,y)$ is considered as a net on the directed set of pairs of positive reals with the product ordering where $(x_{1},x_{2})\leqslant(y_{1},y_{2})$ if and only if $x_{1}\leqslant{y_{1}}$ and $x_{2}\leqslant{y_{2}}$, then the simultaneous limit on this directed set of the net $\dfrac{f(x_{1},x_{2})}{x_{1}\cdot{x_{2}}}$ is its greatest lower bound. This is a generalization of the original statement, where the functions depend on multiple independent real variables, both notions of subadditivity and limit are extended, and the original lemma is a special case for $d=1$. The double limit is also remarkable, because multiple limits need not commute, let alone coincide with a simultaneous limit.

A similar statement for functions defined on $d$-tuples of positive integers (instead of reals) was proved in [1]; see also [10] for an application. The argument presented there, however, relies on a hidden hypothesis of boundedness on compact subsets, which comes for free in the integer setting (where compact subsets are precisely the finite subsets) but must be proved in the new one, and cannot be inferred from boundedness in each variable however given the others (see Example 4.3). By adapting the proof of [6, Theorem 6.4.1] we obtain the following result: componentwise subadditive functions defined on suitable regions of $\mathbb{R}^{d}$ are indeed bounded on compact subsets. For $d=2$ and positive variables the statement goes as follows:

Proposition 1.2**.**

In the hypotheses of Proposition 1.1, the function $f$ is bounded on $[a,b]\times[c,d]$ for every $0<a<b$ and $0<c<d$.

The paper is organized as follows. Section 2 provides the theoretical background. In Section 3 we introduce componentwise subadditivity and explain how it is different from subadditivity in the product semigroup. In Section 4 we adapt the argument from [6, Theorem 6.4.1] to prove that componentwise subadditive functions of $d$ real variables are bounded on compact subsets of $\mathbb{R}^{d}$. In Section 5 we state, prove, and discuss the main theorem: boundedness will have a crucial role in the proof. Section 6 is a discussion on how the beautiful Ornstein-Weiss lemma [11], an important result on subadditive functions defined on finite subsets of groups of a certain class which includes $\mathbb{Z}$ and $\mathbb{R}$, is not an extension of Fekete’s lemma.

2 Background

Throughout the paper, the subsets of $\mathbb{R}^{d}$ and the real-valued functions of real variables are presumed to be Lebesgue measurable. We also let real-valued functions take value either $+\infty$ or $-\infty$, but not both.

We denote by $\mathbb{R}$, $\mathbb{R}_{+}$, and $\mathbb{R}_{-}$ the sets of real numbers, positive real numbers, and negative real numbers, respectively. Similarly, we denote by $\mathbb{Z}$, $\mathbb{Z}_{+}$, and $\mathbb{Z}_{-}$ the sets of integers, positive integers, and negative integers, respectively. All these sets are considered as additive semigroups (groups in the case of $\mathbb{R}$ and $\mathbb{Z}$). If $m$ and $n$ are integers and $m\leqslant{n}$ we denote the slice $\{m,m+1,\ldots,n-1,n\}=\left[m,n\right]\cap\mathbb{Z}$ as $\left[{m}\!:\!{n}\right]$. If $X$ is a set, we denote by $\mathcal{P}\mathcal{F}(X)$ the set of its finite subsets.

If the sets $X$ and $Y$ where the variable $x$ and the expression $E(x)$ take values are irrelevant or clear from the context, we denote by ${x}\mapsto{E(x)}$ the function that associates to each value $\overline{x}$ taken by $x$ the value $E(\overline{x})$. For example, ${x}\mapsto{1}$ is the constant function taking value 1 everywhere.

A directed set is a partially ordered set $\mathcal{U}=(U,\preccurlyeq)$ with the following additional property: for every $x,y\in{U}$ there exists $z\in{U}$ such that $x\preccurlyeq{z}$ and $y\preccurlyeq{z}$. Every totally ordered set is a directed set, and so is the family of decompositions $x=\left\{a=x_{0}<x_{1}<\ldots<x_{n}=b\right\}$ of the compact interval $[a,b]$ with the partial order $x\preccurlyeq{y}$ iff for every $i$ there exists $j$ such that $x_{i}=y_{j}$. A function $f$ defined on $U$ is also called a net on $\mathcal{U}$. If $Y$ is the codomain of $f$, a subnet of $f$ is a net $g:V\to Y$ on a directed set $\mathcal{V}=(V,\leqslant)$ together with a function $\phi:V\to U$ such that:

$f\circ\phi=g$; and 2. 2.

for every $x\in{U}$ there exists $y\in{V}$ such that, if $z\in{V}$ and $z\geqslant{y}$, then $\phi(z)\succcurlyeq{x}$.

For example, a subsequence $\{x_{n_{k}}\}_{k\geqslant 1}$ of a sequence $\{x_{n}\}_{n\geqslant 1}$ of real numbers is a subnet, with $V=U=\mathbb{Z}_{+}$ and $\phi(k)=n_{k}$.

If $\mathcal{U}=(U,\preccurlyeq)$ is a directed set and $f:U\to\mathbb{R}$ is a function, the lower limit and the upper limit of $f$ in $\mathcal{U}$ are the values

[TABLE]

and

[TABLE]

respectively. The following chain of equalities holds:

[TABLE]

Moreover, if $\mathcal{V}=(V,\leqslant)$ is a directed set and $g:V\to\mathbb{R}$ is a subnet of $f$, then:

[TABLE]

If $\liminf_{x\to\mathcal{U}}f(x)=\limsup_{x\to\mathcal{U}}f(x)$, their common value $L$ is called the limit of $f$ in $\mathcal{U}$, and we say that $f$ converges to $L$ in $\mathcal{U}$. This is equivalent to the following: for every $\varepsilon>0$ there exists $x_{\varepsilon}\in U$ such that $|f(x)-L|<\varepsilon$ for every $x\succcurlyeq x_{\varepsilon}$. In this case, every subnet of $f$ also converges to $L$.

The ordered product of a family $\{(X_{i},\leqslant_{i})\}_{i\in I}$ of ordered sets is the ordered set $(X,\leqslant_{\Pi})$ where $X=\prod_{i\in I}X_{i}$ and the product ordering $\leqslant_{\Pi}$ is defined as:

[TABLE]

If each $(X_{i},\leqslant_{i})$ is a directed set, then so is $(X,\leqslant_{\Pi})$. For $d\geqslant 2$ and $w\in\{0,1\}^{d}$ the orthant denoted by $w$ is the directed set $\mathcal{R}_{w}=\prod_{i=1}^{d}(X_{i},\leqslant_{i})$ where $(X_{i},\leqslant_{i})=(\mathbb{R}_{+},\leqslant)$ if $w_{i}=0$ and $(X_{i},\leqslant_{i})=(\mathbb{R}_{-},\geqslant)$ if $w_{i}=1$. For example, $\mathcal{R}_{10}$ is the open second quadrant of the Cartesian plane, with $(x_{1},x_{2})\leqslant(y_{1},y_{2})$ if and only if $x_{1}\geqslant{y_{1}}$ and $x_{2}\leqslant{y_{2}}$. In particular, the main orthant of $\mathbb{R}^{d}$, corresponding to $w=0^{d}$, is $\mathcal{R}_{+}^{d}=\left(\mathbb{R}_{+}^{d},\leqslant_{\Pi}\right)$. Note that, if $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ is a net on $\mathcal{R}_{+}^{d}$ and $\{x_{i,n}\}_{n\geqslant 1}$, $i\in\left[{1}\!:\!{d}\right]$, are sequences of positive reals such that $\lim_{n\to\infty}x_{i,n}=+\infty$ for every $i\in\left[{1}\!:\!{d}\right]$, then $g(n)=f\left(x_{1,n},\ldots,x_{d,n}\right)$ is a subnet of $f$: consequently, if $f$ converges to $L\in\mathbb{R}$ in $\mathcal{R}_{+}^{d}$, then $g(n)$ converges to $L$ for $n\to\infty$.

3 Componentwise subadditivity

In the literature, subadditivity is most often studied in functions of a single variable, which sometimes may be vector rather than scalar. But in some cases, it is of interest to consider functions of $d$ independent variables, which are subadditive when considered as functions of only one of those, but however given the remaining ones.

Definition 3.1**.**

Let $S_{1},\ldots,S_{d}$ be semigroups, let $S=\prod_{i=1}^{d}S_{i}$, and let $f:S\to\mathbb{R}$. Given $i\in\left[{1}\!:\!{d}\right]$, we say that $f$ is subadditive in $x_{i}$ independently of the other variables if, however given $x_{j}\in S_{j}$ for every $j\in\left[{1}\!:\!{d}\right]\setminus\{i\}$, the function ${x_{i}}\mapsto{f(x_{1},\ldots,x_{i},\ldots,x_{d})}$ is subadditive on $S_{i}$. We say that $f$ is componentwise subadditive if it is subadditive in each variable independently of the others.

Example 3.2**.**

If $f_{1}:S_{1}\to\mathbb{R}$ and $f_{2}:S_{2}\to\mathbb{R}$ are both subadditive and nonnegative, then $f:S_{1}\times S_{2}\to\mathbb{R}$ defined by $f(x_{1},x_{2})=f_{1}(x_{1})\cdot f_{2}(x_{2})$ is componentwise subadditive.

If one between $f_{1}$ and $f_{2}$ takes negative values, then $f$ might not be componentwise subadditive. For example, $f(x_{1})=-x_{1}$ is subadditive on $\mathbb{R}_{+}$, because it is linear, and $f_{2}(x_{2})=\sqrt{x_{2}}$ is also subadditive on $\mathbb{R}_{+}$, because it is nondecreasing and $x_{2}+y_{2}<\left(\sqrt{x_{2}}+\sqrt{y_{2}}\right)^{2}$ for every $x_{2},y_{2}>0$; but for any fixed $x_{1}>0$, the function ${x_{2}}\mapsto{-x_{1}\sqrt{x_{2}}}$ is not subadditive on $\mathbb{R}_{+}$.

Example 3.3**.**

(cf. [1, Section 3])

Let $d$ be a positive integer and let $A$ be a finite set with $a\geqslant 2$ elements, considered as a discrete space. The translation by $v\in\mathbb{Z}^{d}$ is the function $\sigma_{v}:A^{\mathbb{Z}^{d}}\to A^{\mathbb{Z}^{d}}$ defined by $\sigma_{v}(c)(x)=c(x+v)$ for every $x\in\mathbb{Z}^{d}$. A $d$-dimensional subshift on $A$ is a subset $X$ of $A^{\mathbb{Z}^{d}}$ which is closed in the product topology and invariant by translation, that is, if $c\in{X}$, then $\sigma_{v}(c)\in{X}$ for every $v\in\mathbb{Z}^{d}$. Given $d$ positive integers $n_{1},\ldots,n_{d}$, an allowed pattern of sides $n_{1},\ldots,n_{d}$ for $X$ is a function $p:\prod_{i=1}^{d}\left[{1}\!:\!{n_{i}}\right]\to A$ such that there exists $c\in{X}$ for which the restriction of $c$ to $\prod_{i=1}^{d}\left[{1}\!:\!{n_{i}}\right]$ coincides with $p$. Let $\mathcal{A}_{X}(n_{1},\ldots,n_{d})$ be the number of allowed patterns for $X$ of sides $n_{1},\ldots,n_{d}$: then

[TABLE]

is componentwise subadditive, because every allowed pattern of sides $n_{1}+m_{1},n_{2},\ldots,n_{d}$ can be obtained by joining an allowed pattern of sides $n_{1},n_{2},$ $\ldots,n_{d}$ with an allowed pattern of sides $m_{1},n_{2},\ldots,n_{d}$, but joining two such allowed patterns does not necessarily produce an allowed pattern; similarly for the other $d-1$ coordinates. This works because $X$ is invariant by translations.

Componentwise subadditivity is very different from subadditivity with respect to the operation of the product semigroup. Already with $d=2$, if $f:S_{1}\times S_{2}\to\mathbb{R}$ is subadditive, then for every $x_{1},y_{1}\in S_{1}$ and $x_{2},y_{2}\in S_{2}$ we have:

[TABLE]

while if $f$ is componentwise subadditive, then for every $x_{1},y_{1}\in S_{1}$ and $x_{2},y_{2}\in S_{2}$ we have the more complex upper bound:

[TABLE]

If $f$ is nonnegative, then (8) implies (9), which however is weaker than the conditions of Definition 3.1; if $f$ is nonpositive, then (9) implies (8). In general, however, neither implies the other.

Example 3.4**.**

By our discussion in Example 3.2, the function $f(x_{1},x_{2})=\sqrt{x_{1}x_{2}}$ is componentwise subadditive on $\mathbb{R}_{+}^{2}$. However, $f$ is not subadditive, because $f(3,3)=3>2\sqrt{2}=f(1,2)+f(2,1)$.

Example 3.5**.**

The function (7) of Example 3.3 is not, in general, subadditive. For example, for $d=2$ and $X=A^{\mathbb{Z}^{2}}$ every pattern is allowed, so $f(n_{1},n_{2})=n_{1}n_{2}$: but if $n_{1}$, $n_{2}$, $m_{1}$, and $m_{2}$ are all positive, then $(n_{1}+m_{1})(n_{2}+m_{2})>n_{1}n_{2}+m_{1}m_{2}$.

Although componentwise subadditivity is very different from subadditivity in the product semigroup, Fekete’s lemma can tell us something important for the case of positive integer or real variables.

Lemma 3.6**.**

Let $S=\prod_{i=1}^{d}S_{i}$ with each $S_{i}$ being either $\mathbb{R}_{+}$ or $\mathbb{Z}_{+}$, and let $f:S^{d}\to\mathbb{R}\cup\{-\infty\}$ be componentwise subadditive. Having fixed $k\in\left[{1}\!:\!{d-1}\right]$, let $i,j_{1},\ldots,j_{k}\in\left[{1}\!:\!{d}\right]$ be pairwise different. However fixed the values of the remaining variables, the function $h:S_{i}\to\mathbb{R}\cup\{-\infty\}$ defined by the multiple limit:

[TABLE]

is subadditive.

Proof.

It is sufficient to prove the thesis for $k=1$; the general case follows by repeated application. To simplify notation, let $j=j_{1}$. Fix the values of $x_{s}$ for $s\in\left[{1}\!:\!{d}\right]\setminus\{i,j\}$. By hypothesis, for every $x_{i}\in{S_{i}}$ the function ${x_{j}}\mapsto{f(x_{1},\ldots,x_{d})}$ is subadditive, so by Fekete’s lemma $h(x_{i})=\lim_{x_{j}\to\infty}\frac{f(x_{1},\ldots,x_{d})}{x_{j}}$ exists. But for every $x_{i},x_{i}^{\prime},x_{j}>0$ it is:

[TABLE]

so it must be $h(x_{1}+x_{1}^{\prime})\leqslant h(x_{1})+h(x_{1}^{\prime})$ too. Note that it is crucial for the proof that $x_{j}>0$. ∎

The following observation is crucial for the next sections.

Proposition 3.7**.**

Let $w=w_{1}\ldots w_{d}$ be a binary word of length $d$ and let $f:\mathbb{R}_{w}\to\mathbb{R}$. For every $i\in\left[{1}\!:\!{d}\right]$ let $x_{w,i}=(-1)^{w_{i}}x_{i}\in\mathbb{R}_{+}$, and let $f_{w}:\mathbb{R}_{+}^{d}\to\mathbb{R}$ be defined by $f_{w}(x_{w,1},\ldots,x_{w,d})=f(x_{1},\ldots,x_{d})$. The following are equivalent:

$f(x_{1},\ldots,x_{d})$* is componentwise subadditive in $\mathbb{R}_{w}$.* 2. 2.

$f_{w}(x_{w,1},\ldots,x_{w,d})$* is componentwise subadditive in $\mathbb{R}_{+}^{d}$.*

The same holds if $\mathbb{R}_{w}$ and $\mathbb{R}_{+}^{d}$ are replaced with $\mathbb{Z}_{w}=\mathbb{R}_{w}\cap\mathbb{Z}^{d}$ and $\mathbb{Z}_{+}^{d}$, respectively.

4 Componentwise subadditive functions of $d$ real variables are bounded on compacts

In [1] we prove the following:

Proposition 4.1** (Fekete’s lemma in $\mathbb{Z}_{+}^{d}$; [1, Theorem 1]).**

Let $\mathcal{U}=\left(\mathbb{Z}_{+}^{d},\leqslant_{\Pi}\right)$ and let $f:\mathbb{Z}_{+}^{d}\to\mathbb{R}$ be componentwise subadditive. Then:

[TABLE]

Example 4.2**.**

With the notation of Example 3.3 and $\mathcal{U}$ as in Proposition 4.1, the value:

[TABLE]

is well defined, and is called the entropy of the subshift $X$. For $d=1$ this coincides with [9, Definition 4.1.1].

We try to reuse the argument from [1] to prove Proposition 1.1. Fix $s,t>0$. Every $x>0$ large enough has a unique writing $x=qs+r$ with $q$ positive integer and $r\in\left[s,2s\right)$, and every $y>0$ large enough has a unique writing $y=mt+p$ with $m$ positive integer and $p\in\left[t,2t\right)$. By subadditivity,

[TABLE]

Consider the four summands on the right-hand side of the last inequality. By construction, $\lim_{x\to+\infty}q/x=1/s$ and $\lim_{y\to+\infty}m/y=1/t$: therefore, the first summand converges to $f(s,t)/st$ for $(x,y)\to\mathcal{R}_{+}^{2}$.

Now, by [6, Theorem 6.4.1], a subadditive function of one positive real variable is bounded in every compact subset of $\mathbb{R}_{+}$. Then ${p}\mapsto{f(s,p)}$ is bounded on $\left[t,2t\right]$ and ${r}\mapsto{f(r,t)}$ is bounded on $\left[s,2s\right]$: consequently, the second and third summand vanish for $(x,y)\to\mathcal{R}_{+}^{2}$.

But the fourth summand presents a problem. What we know, is that ${x}\mapsto{f(x,y)}$ is bounded in $[s,2s]$ for every $y\in[t,2t]$, and ${y}\mapsto{f(s,y)}$ is bounded in $[t,2t]$ for every $x\in[s,2s]$. This is, in general, strictly less than $f$ being bounded in $[s,2s]\times[t,2t]$: which is what we actually need to show that the fourth summand vanishes when $x$ and $y$ both grow arbitrarily large!

Example 4.3** (suggested by Arthur Rubin).**

Let $h:\mathbb{R}_{+}\to\mathbb{R}$ be such that $h(t)$ is the denominator of the representation of $t$ as an irreducible fraction if $t$ is rational, and 0 if $t$ is irrational. Then $f:\mathbb{R}_{+}^{2}\to\mathbb{R}$ defined by $f(x,y)=\min(h(x),h(y))$ satisfies the following conditions:

for every $x\in[1,2]$, the function ${y}\mapsto{f(x,y)}$ is bounded in $[1,2]$; 2. 2.

for every $y\in[1,2]$, the function ${x}\mapsto{f(x,y)}$ is bounded in $[1,2]$.

However, $f$ is not bounded in $[1,2]\times[1,2]$, because $f(1+1/n,1+1/n)=n$ for every $n\in\mathbb{Z}_{+}$. On the other hand, $h(4)=1$ and $h(\pi)=h(4-\pi)=0$, so $f$ is neither subadditive nor componentwise subadditive in $\mathbb{R}_{+}^{2}$.

We could overcome this issue if a result of boundedness such as the one in [6, Theorem 6.4.1] held for componentwise subadditive functions. Luckily, it is so, and we can follow the same idea of Hille’s proof. Given $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ and $t_{1},\ldots,t_{d}\in\mathbb{R}_{+}$, let:

[TABLE]

Under our hypothesis that $f$ is measurable, so is (13).

The next statement is the cornerstone of our argument. For Lemma 4.4 and Theorem 4.5, the symbol $\mu$ and the word “measure” denote the $d$-dimensional Lebesgue measure.

Lemma 4.4**.**

Let $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ be componentwise subadditive. Then for every $t_{1},\ldots,t_{d}\in\mathbb{R}_{+}$,

[TABLE]

Proof.

Call $V$ the set on the left-hand side of (14). For every $i\in\left[{1}\!:\!{d}\right]$, given $x_{i}\in(0,t_{i})$, let $y_{i}^{(0)}=x_{i}$ and $y_{i}^{(1)}=t_{i}-x_{i}$. Then for every $w\in\{0,1\}^{d}$ the transformation

[TABLE]

is a measure-preserving continuous involution, hence the set:

[TABLE]

is measurable and satisfies $\mu(V_{w})=\mu(V)$. Note that $V=V_{0^{d}}$.

By repeatedly applying subadditivity, once in each variable, we arrive at:

[TABLE]

For example, for $d=2$ we have:

[TABLE]

For (15) to hold, at least one of the $2^{d}$ summands on the right-hand side must be no smaller than $\dfrac{f(t_{1},\ldots,t_{d})}{2^{d}}$. Then $\bigcup_{w\in\{0,1\}^{d}}V_{w}=\prod_{i=1}^{d}(0,t_{i})$, so:

[TABLE]

∎

From Lemma 4.4 follows:

Theorem 4.5**.**

Let $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ be componentwise subadditive. Then $f$ is bounded in every compact subset of $\mathbb{R}_{+}^{d}$.

Proof.

It is sufficient to prove the thesis for every compact hypercube of the form $H=[a,b]^{d}$ with $0<a<b$. We proceed by contradiction, following the argument from [6, Theorem 6.4.1].

First, suppose that $f$ is unbounded from above in $H$. Then for every $n\geqslant 1$ and $i\in\left[{1}\!:\!{d}\right]$ there exists $x_{i,n}\in[a,b]$ such that $f(x_{1,n},\ldots,x_{d,n})\geqslant 2^{d}n$. Let $W_{t_{1},\ldots,t_{d}}$ be the set in (14). By construction, for every $n\geqslant 1$ we have

[TABLE]

and by Lemma 4.4,

[TABLE]

Now, the sets $V_{b,\ldots,b,n}$ are measurable and form a nonincreasing sequence, so $V=\bigcap_{n\geqslant 0}V_{b,\ldots,b,n}$ is measurable and $\mu(V)\geqslant\left(a/2\right)^{d}$: in particular, $V$ cannot be empty. But for $(x_{1},\ldots,x_{d})\in V$ it must be $f(x_{1},\ldots,x_{d})\geqslant n$ for every $n\geqslant 1$: which is impossible.

Next, suppose that $f$ is unbounded from below in $H$. Then for every $n\geqslant 1$ and $i\in\left[{1}\!:\!{d}\right]$ there exists $x_{i,n}\in[a,b]$ such that $f(x_{1,n},\ldots,x_{d,n})\leqslant-n$: we may assume that $\lim_{n\to\infty}x_{i,n}=x_{i}\in[a,b]$ exists for every $i\in\left[{1}\!:\!{d}\right]$. Let $s=\min(a,1)$, $t=b+4$, and $J=[s,t]^{d}$: then every point $(z_{1},\ldots,z_{d})$ where each $z_{i}$ belongs to either $[a,b]$ or $[1,4]$ belongs to $J$. Let now $y_{i}\in[1,4]$ for every $i\in\left[{1}\!:\!{d}\right]$ and

[TABLE]

which is a real number because of the previous point. By applying subadditivity in each variable, for such $y_{1},\ldots,y_{d}$ and $n$ we obtain

[TABLE]

because $-n$ is an upper bound for $f(x_{1,n},\ldots,x_{d,n})$ and $M$ is an upper bound for the other $2^{d}-1$ summands, For example, for $d=2$ we have:

[TABLE]

But for every $n$ such that $|x_{i,n}-x_{i}|\leqslant 1$ it is $[x_{i}+2,x_{i}+3]\subseteq[x_{i,n}+1,x_{i,n}+4]$: calling

[TABLE]

for every $n$ large enough every element of $K$ can be written in the form $(y_{1}+x_{1,n},\ldots,y_{d}+x_{d,n})$ for suitable $y_{1},\ldots,y_{d}\in[1,4]$. For every $(z_{1},\ldots,z_{d})\in K$ it must then be $f(z_{1},\ldots,z_{d})\leqslant(2^{d}-1)M-n$ for every $n$ large enough: which is impossible. ∎

If $f:\mathbb{R}_{w}\to\mathbb{R}$ is bounded on the compact subsets of $\mathbb{R}_{w}$, then $f_{w}$ as defined in Proposition 3.7 is bounded on the compact subsets of $\mathbb{R}_{+}$; and vice versa. From Theorem 4.5 and Proposition 3.7 follows:

Corollary 4.6**.**

Let $w\in\{0,1\}^{d}$ and let $f:\mathbb{R}_{w}\to\mathbb{R}$ be componentwise subadditive. Then $f$ is bounded in every compact subset of $\mathbb{R}_{w}$.

In turn, Corollary 4.6 allows us to prove:

Theorem 4.7**.**

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be componentwise subadditive. Then $f$ is bounded in every compact subset of $\mathbb{R}^{d}$.

Proof.

It is sufficient to show finitely many open sets $V_{1},\ldots,V_{n}$ such that $f$ is bounded on the compacts of each $V_{i}$ and:

[TABLE]

We give the argument for $d=3$: the ideas for arbitrary $d\geqslant 1$ are similar. Let $I=[-1/2,1/2]$ and $U=[-3/2,-1/2]\cup[1/2,3/2]$.

We start by proving that $f$ is bounded in every compact subset of the open set

[TABLE]

where $D_{00}=\{(x,y,z)\in\mathbb{R}^{3}\mid x>0,y>0,z=0\}$ is the first quadrant of the $XY$ plane. To do this, we only need to show that $f$ is bounded in every set of the form $H=[a,b]\times[a,b]\times I$. Let $V=[a,b]\times[a,b]\times U$: if $(x,y,z)\in{H}$, then $(x,y,z-1)$ and $(x,y,z+1)$ are both in $V$. Let $T$ and $t$ be an upper bound and a lower bound for $f$ in $V$, respectively: then for every $(x,y,z)\in{H}$,

[TABLE]

and

[TABLE]

By similar arguments, $f$ is bounded in every compact subset of every subset of $\mathbb{R}^{3}$ which is the union of two adjacent orthants and the corresponding “quadrant”. As for each open orthant there are three which border it by one “quadrant”, there are $\dfrac{8\cdot 3}{2}=12$ such subsets.

We now show that $f$ is bounded in every compact subset of the open “upper demispace” $Z_{0}=\{(x,y,z)\in\mathbb{R}^{3}\mid z>0\}$. To do so, it will suffice to show that $f$ is bounded in every set of the form $L=I\times I\times[a,b]$ with $0<a<b$. Let $W=U\times U\times[a,b]$ and let $S$ and $s$ be an upper bound for $f$ in $W$, respectively: then for every $(x,y,z)\in{L}$,

[TABLE]

and

[TABLE]

Similarly, $f$ is bounded in each of the other five open “demispaces”.

To conclude the proof, we only need to show that $f$ is bounded in $K=I\times I\times I$. Let $E=U\times U\times U$ and let $M$ and $m$ be an upper bound and a lower bound for $f$ in $E$, respectively: then for every $(x,y,z)\in{K}$,

[TABLE]

and

[TABLE]

∎

Note that the argument of Lemma 4.4 also works if $f$ is subadditive, rather than componentwise subadditive. In this case, however, the denominator in (14) and in the thesis is 2 rather than $2^{d}$. A more complex variant of it can then be stated, where $f$ is a function of $k$ variables $x_{i}$, each taking values in an orthant of $\mathbb{R}_{+}^{d_{i}}$: and the denominator would then be $2^{k}$. From this, a generalization of Theorem 4.5 to the case of componentwise functions of $k$ variables, the $i$th of which takes values in $\mathbb{R}_{+}^{d_{i}}$, can be derived.

5 Fekete’s lemma for componentwise subadditive functions of $d$ real variables

We can now state and prove the main result of this paper.

Theorem 5.1** (Fekete’s lemma in $\mathbb{R}_{+}^{d}$).**

Let $d\geqslant 1$ and let $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ be componentwise subadditive. Then:

[TABLE]

which can be $-\infty$. In addition, for every permutation $\sigma$ of $\left[{1}\!:\!{d}\right]$,

[TABLE]

regardless of any of the limits being finite or (negatively) infinite.

The proof of (16) is similar to that of [1, Theorem 1], with an important change; for convenience of the reader, we report it entirely. The proof of (17) relies on (16), the original Fekete’s lemma, and the following Lemma 5.2; the argument remains valid for the case of positive integer variables.

Lemma 5.2**.**

Let $u$ be a real-valued function depending on $d\geqslant 1$ variables $x_{i}$, no matter of what type. Then for every permutation $\sigma$ of $\left[{1}\!:\!{d}\right]$,

[TABLE]

Sketch of proof..

Let $L$ and $R$ be the left- and right-hand side of (18), respectively. It is easy to see that $L\geqslant{R}$. Let now $\delta>0$: there exist $z_{1},\ldots,z_{d}\in\mathbb{R}_{+}$ such that $u(z_{1},\ldots,z_{d})<R+\delta$, so a fortiori $L<R+\delta$ too; as $\delta>0$ is arbitrary, $L\leqslant{R}$. ∎

Proof of Theorem 5.1.

For every $i\in\{1,\ldots,d\}$ and $x_{1},\ldots,x_{d}\in\mathbb{R}_{+},$ if $x_{i}=qt+r$ with $q\in\mathbb{Z}_{+}$ and $r,t\in\mathbb{R}_{+}$, then:

[TABLE]

Fix $t_{1},\ldots,t_{d}\in\mathbb{R}_{+}$. For every $i\in\left[{1}\!:\!{d}\right]$ and $x_{i}\geqslant{2t_{i}}$ there exist unique $q_{i}\in\mathbb{Z}_{+}$ and $r_{i}\in\left[t_{i},2t_{i}\right)$ such that $x_{i}=q_{i}t_{i}+r_{i}$. For every $i\in\left[{1}\!:\!{d}\right]$ let $y_{i}^{(0)}=r_{i}$ and $y_{i}^{(1)}=t_{i}$: by repeatedly applying subadditivity, once per each variable, we find:

[TABLE]

Now, on the right-hand side of (19), each occurrence of $f$ has $k$ arguments chosen from the $t_{i}$’s and $d-k$ chosen from the $r_{i}$’s, is multiplied by the $q_{i}$’s corresponding to the $t_{i}$’s, and is bounded from above by the constant

[TABLE]

which exists because of Theorem 4.5. Such boundedness is crucial for the proof, and was ensured for free in the case of positive integer variables from [1], but had to be proved for positive real variables.

By dividing both sides of (19) by $x_{1}\cdots{x_{d}}$ we get:

[TABLE]

where $1^{d}$ is the binary word of length $d$ where all the bits are 1.

By construction, $\lim_{x_{i}\to\infty}q_{i}/x_{i}=1/t_{i}$. Given $\varepsilon>0$, choose $x_{1}^{(\varepsilon)},\ldots,x_{d}^{(\varepsilon)}\in\mathbb{R}_{+}$ such that, if $x_{i}\geqslant x_{i}^{(\varepsilon)}$ for each $i\in\left[{1}\!:\!{d}\right]$, then the following hold:

$\dfrac{f(x_{1},\ldots,x_{d})}{x_{1}\cdots x_{d}}<\dfrac{f(t_{1},\ldots,t_{d})}{t_{1}\cdots t_{d}}+\dfrac{\varepsilon}{2^{d}}$; 2. 2.

$\dfrac{q_{1}^{w_{1}}\cdots q_{d}^{w_{d}}}{x_{1}\cdots x_{d}}<\dfrac{\varepsilon}{M\cdot 2^{d}}$ for every $w\in\{0,1\}^{d}\setminus\{1^{d}\}$.

This is possible because if $w\neq 1^{d}$, then at least one of the $q_{i}^{w_{i}}$ equals 1. For such $x_{1},\ldots,x_{d}$ it is:

[TABLE]

As $\varepsilon>0$ is arbitrary, it must be:

[TABLE]

But the $t_{i}$’s are also arbitrary, hence:

[TABLE]

which yields (16).

Now, by Lemma 3.6, for every choice of $i,j_{1},\ldots,j_{k}\in\left[{1}\!:\!{d}\right]$ all different, and however fixed the remaining variables, the function (10) is subadditive. Then (17) follows from Lemma 5.2 by repeated application of the original Fekete’s lemma:

[TABLE]

∎

From Theorem 5.1 and Proposition 3.7 follows:

Theorem 5.3**.**

Let $d\geqslant 1$, let $w,w^{\prime}\in\{0,1\}^{d}$ and let $f:\mathbb{R}_{w}\to\mathbb{R}$ be componentwise subadditive.

If $w$ contains evenly many 1s, then:

[TABLE]

is not $+\infty$, but can be $-\infty$. 2. 2.

If $w$ contains oddly many 1s, then:

[TABLE]

is not $-\infty$, but can be $+\infty$. 3. 3.

Suppose now $w$ contains evenly many 1s, $w^{\prime}$ differs from $w$ in exactly one coordinate, and $f$ is defined and componentwise subadditive in $\mathbb{R}_{w}\cup\mathbb{R}_{w^{\prime}}\cup U_{w,w^{\prime}}$, where:

[TABLE]

is the boundary between $\mathbb{R}_{w}$ and $\mathbb{R}_{w^{\prime}}$. Then:

[TABLE]

consequently, both limits are finite.

For $d=1$ we recover [6, Theorem 6.6.1]. To prove Theorem 5.3, we make use of the following result, whose proof we leave to the reader.

Lemma 5.4**.**

Let $S$ be a semigroup and $f:S\to\mathbb{R}$ be a subadditive function. If $S$ is a monoid with identity $e$, then $f(e)\geqslant 0$. If, in addition, $S$ is a group, then $f(x)+f(x^{-1})\geqslant 0$ for every $x\in{S}$.

Proof of Theorem 5.3.

For $(x_{1},\ldots,x_{d})\in\mathcal{R}_{w}$ and $i\in\left[{1}\!:\!{d}\right]$ let $x_{w,i}$ and $f_{w}$ be defined as in Proposition 3.7. If $w$ contains evenly many 1s, then $x_{1}\cdots x_{d}=x_{w,1}\cdots x_{w,d}$ and:

[TABLE]

If $w$ contains oddly many 1s, then $x_{1}\cdots x_{d}=-x_{w,1}\cdots x_{w,d}$ and:

[TABLE]

Suppose now $w$ has evenly many 1s and $w^{\prime}$ differs from $w$ only in component $i$, and $f$ is defined and componentwise subadditive in $\mathbb{R}_{w}\cup\mathbb{R}_{w^{\prime}}\cup U_{w,w^{\prime}}$. Then for every $x_{1},\ldots,x_{i-1},x_{i+1},\ldots x_{d}$ the function ${x_{i}}\mapsto{f(x_{1},\ldots,x_{i},\ldots,x_{d})}$ is subadditive on $\mathbb{R}$: by Lemma 5.4, for every $x_{i}\in\mathbb{R}$,

[TABLE]

Then:

[TABLE]

is nonnegative. The last passage is valid because the two limits on the second line are either finite or $-\infty$. ∎

As every subnet of a convergent net converges to the same limit, we get:

Corollary 5.5**.**

Let $f:\mathbb{R}_{+}^{d}\to\mathbb{R}$ be componentwise subadditive and let $T$ be either $\mathbb{R}_{+}$ or $\mathbb{Z}_{+}$. For every $i\in\left[{1}\!:\!{d}\right]$ let $x_{i}(t):T\to\mathbb{R}_{+}$ satisfy $\lim_{t\to+\infty}x_{i}(t)=+\infty$. Then:

[TABLE]

and also

[TABLE]

In particular,

[TABLE]

Sketch of proof.

We only remark that (25) follows from (16) and:

[TABLE]

∎

Note that, in general, even if $f$ is componentwise subadditive on $\mathbb{R}_{+}^{d}$, ${t}\mapsto{f(t,\ldots,t)}$ is not subadditive on $\mathbb{R}_{+}$: a simple example is $f(x_{1},x_{2})=x_{1}\cdot x_{2}$. This provides further evidence that Theorems 5.1 and 5.3 are not special cases of [6, Theorem 6.6.1].

A real-valued function defined on a semigroup $(S,\cdot)$ is superadditive if it satisfies $f(x\cdot y)\geqslant f(x)+f(y)$ for every $x,y\in{S}$. As $f$ is superadditive if and only if $-f$ is subadditive, an analogue of Theorem 5.1 holds for componentwise superadditive functions, provided one swaps the roles of $\inf$ and $\sup$ and those of $-\infty$ and $+\infty$. If $f$ is superadditive in some variables and subadditive in other variables, however, Theorem 5.1 does not hold.

Example 5.6**.**

The function $f:\mathbb{R}_{+}^{2}\to\mathbb{R}$ defined by $f(x_{1},x_{2})=x_{1}^{2}\sqrt{x_{2}}$ is superadditive in $x_{1}$ and subadditive in $x_{2}$, and $f(x_{1},x_{2})/x_{1}x_{2}=x_{1}/\sqrt{x_{2}}$. But $\lim_{(x_{1},x_{2})\to\mathcal{R}_{+}^{2}}f(x_{1},x_{2})/x_{1}x_{2}$ does not exist, because for every $y,R>0$ there exist $x_{1},x_{2}>R$ such that $x_{1}/\sqrt{x_{2}}=y$. Also, $\lim_{x_{1}\to\infty}\lim_{x_{2}\to\infty}\dfrac{f(x_{1},x_{2})}{x_{1}\cdot x_{2}}=0$ but $\lim_{x_{2}\to\infty}\lim_{x_{1}\to\infty}\dfrac{f(x_{1},x_{2})}{x_{1}\cdot x_{2}}=+\infty$.

As a final remark for this section, the following statement appears in the literature as an extension to arbitrary dimension of [6, Theorem 6.1.1]:

Proposition 5.7** (cf. [8, Theorem 16.2.9]).**

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be subadditive in the variable $\mathbf{x}\in\mathbb{R}^{d}$. Then for every $\mathbf{x}\in\mathbb{R}^{d}$ the following limit exists:

[TABLE]

This, however, is not so much an extension than a corollary. If $f:\mathbb{R}^{d}\to\mathbb{R}$ satisfies $f(\mathbf{x}+\mathbf{y})\leqslant f(\mathbf{x})+f(\mathbf{y})$ for every $\mathbf{x},\mathbf{y}\in\mathbb{R}^{d}$, then obviously $g_{\mathbf{x}}(t)=f(t\mathbf{x})$ satisfies $g_{\mathbf{x}}(s+t)\leqslant g_{\mathbf{x}}(s)+g_{\mathbf{x}}(t)$ for every $s,t>0$: and $L_{\mathbf{x}}$ is simply the limit of $g_{\mathbf{x}}(t)/t$ according to [6, Theorem 6.1.1]. On the other hand, Theorem 5.3 is an extension.

6 A comparison with the Ornstein-Weiss lemma

A group $G$ is amenable if there exist a directed set $\mathcal{U}=(U,\preccurlyeq)$ and a net $\{F_{x}\}_{x\in U}$ of finite nonempty subsets of $G$ such that:

[TABLE]

A net such as in (27) is called a (left) Følner net on the group $G$, from the Danish mathematician Erling Følner who introduced them in [4]. Every abelian group is amenable: for a proof, see [2, Chapter 4].

Proposition 6.1** (Ornstein-Weiss lemma; cf. [11]).**

Let $G$ be an amenable group and let $f:\mathcal{P}\mathcal{F}(G)\to\mathbb{R}$ be a function which:

is subadditive with respect to set union, that is, $f(A\cup B)\leqslant f(A)+f(B)$ for every $A,B\in\mathcal{P}\mathcal{F}(G)$; and 2. 2.

satisfies $f(A)=f(gA)$ for every $A\in\mathcal{P}\mathcal{F}(G)$ and $g\in{G}$.

Then for every directed set $\mathcal{U}=(U,\preccurlyeq)$ and every left Følner net $\mathcal{F}=\{F_{x}\}_{x\in U}$ on $G$,

[TABLE]

exists, and does not depend on the choice of $\mathcal{U}$ and $\mathcal{F}$.

The Ornstein-Weiss lemma says that, for “well behaving” functions on amenable groups, a notion of asymptotic average is well defined. A detailed proof of Proposition 6.1 is given by F. Krieger in [7].

Example 6.2**.**

Let $G$ be an amenable group and let $A$ be a finite set with $a\geqslant 2$ elements. The shift by $g\in{G}$ is the function $\sigma_{g}:A^{G}\to A^{G}$ defined by $\sigma_{g}(c)(x)=c(g\cdot x)$ for every $c\in{A^{G}}$ and $x\in{G}$. The notions of subshift and of allowed pattern with support $S\in\mathcal{P}\mathcal{F}(G)$ are extended naturally from those of Example 3.3. Calling $\mathcal{A}_{X}(S)$ the number of allowed patterns for $X$ with support $S$, and convening that the unique empty pattern $e:\emptyset\to{A}$ appears in every configuration, we have for every $S,T\in\mathcal{P}\mathcal{F}(G)$:

[TABLE]

Indeed, every allowed pattern on $S$ (resp., $T\setminus{S}$) can be extended to at least one allowed pattern on $S\cup{T}$ (resp., $T$) but joining an allowed pattern over $S$ and an allowed pattern over $T\setminus{S}$ does not necessarily yield an allowed pattern on $S\cup{T}$. Hence, $f(S)=\log_{a}\mathcal{A}_{X}(S)$ is subadditive on $\mathcal{P}\mathcal{F}(G)$, and clearly satisfies $f(gS)=f(S)$ for every $g\in{G}$ and $S\in\mathcal{P}\mathcal{F}(G)$. The entropy of $X$ can then be defined as:

[TABLE]

where $\mathcal{U}=(U,\preccurlyeq)$ is an arbitrary directed set and $\{F_{x}\}_{x\in U}$ is an arbitrary Følner net on $G$.

As the sets $E_{x_{1},\ldots,x_{d}}=\prod_{i=1}^{d}\left[{1}\!:\!{x_{i}}\right]$ with $x_{1},\ldots,x_{d}\in\mathbb{Z}_{+}$ constitute a Følner net on $\mathbb{Z}^{d}$, defining the entropy of a $d$-dimensional subshift according to either Example 4.2 or Example 6.2 yields the same result. Nevertheless, the Ornstein-Weiss lemma does not generalize Fekete’s lemma, nor it is possible to prove the latter from the former, as the limit (28) is only ensured to exist, not to coincide with any specific value. In addition, even if $f:\mathbb{Z}\to\mathbb{R}$ is subadditive, the “natural” conversion

[TABLE]

where $|A|$ is the number of elements of $A$, is invariant by translations, but needs not be subadditive on $\mathcal{P}\mathcal{F}(\mathbb{Z})$, the main reason being that $|A\cup{B}|$ needs not equal $|A|+|B|$. Moreover, while invariance by translations is essential in the Ornstein-Weiss lemma, a translate of a subadditive function needs not be subadditive.

Example 6.3**.**

The function $f(n)=n\!\!\mod 2$ is easily seen to be subadditive on $\mathbb{Z}$. But the function $g$ defined from $f$ by (30) is not subadditive on $\mathcal{P}\mathcal{F}(\mathbb{Z})$, because if $U=\{1,2\}$ and $V=\{2,3\}$, then $g(U\cup{V})=1$ and $g(U)=g(V)=0$. Note that $h(n)=f(n+1)$ is not subadditive, because $h(1)=0$ but $h(2)=1$.

7 Conclusions

We have discussed an extension of the notion of subadditivity in the case of many independent variables. In this context, we have proved a nontrivial extension of the classical Fekete’s lemma to the case of functions of $d\geqslant 1$ real variables, which recovers the original statement for $d=1$, and which is more general than other extensions already present in the literature. While doing so, we have also proved that these componentwise subadditive functions satisfy the important property of being bounded on compact subsets, the case $d=1$ being already known from the literature.

We believe that our results can be of interest for researchers in economics, theory of dynamical systems, and mathematical analysis.