arXiv:1904.10507v4 [math.CA] 3 Feb 2021 o every for ois nra nlss uadtv ucin r uhmo much are functions subadditive economics, analysis, [9], real theory In give hav information a functions including torics. of Subadditive fields subset alphabet. finite many an a over in word a a of (small of cardinality number length the real a it); of than ceiling the smaller number; complex a of value 11 21 aln,Estonia. Tallinn, 12618 21/1, elvle function real-valued A Introduction 1 ntttoa eerhgatn.IT31 n yteEstoni the by and IUT33-13 no. PRG1210. grant research institutional ∗ † hsrsac a upre yteEtna iityo Educ of Ministry Estonian the by supported was research This eateto otaeSine aln nvriyo Tech of University Tallinn Science, Software of Department eeeslmafrcmoetiesubadditive componentwise for lemma Fekete’s e od:sbdiiiy utvraeaayi,smlaeu limit. simultaneous analysis, multivariate 00A05. subadditivity, 26B35, words: 39B62, Key E. 2010: of Classification 6 Subject Chapter Mathematics in those on prop- expand the textbook. arguments sub with 1948 The bounded Hille’s functions and th domain. the closed by every their that in paper of prove bounded previous are we a above so, in mentioned erty result doing subaddit a While for and variable case variable, each author. classical real in the one subadditive both of are functions extends which This variables singularly. real taken several of tions epoea nlgeo eeessbdiiiylmafrfunc- for lemma subadditivity Fekete’s of analogue an prove We ,y x, ucin ftoo oera variables real more or two of functions ∈ S xmlso uadtv ucin nld h absolute the include functions subadditive of Examples . f endo eiru ( semigroup a on defined [email protected] f ivoCapobianco Silvio ( x · y Abstract ) 6 f 1 ( x + ) f ( y ∗† (1) ) S, nRsac oni grant Council Research an · is ) ooy kdei tee Akadeemia nology. subadditive to n Research and ation e;adthe and set; n s nee not integer est n combina- and applications e einteresting re set ive e if than additive ones: for example, every additive, Lebesgue measurable func- tion 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 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 mathe- matical 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 S is a product semigroup, we can also consider the 1 × 2 case of a function which is subadditive in each variable, however given the other. That is, instead of requiring f(x1y1,x2y2) 6 f(x1,x2)+ f(y1,y2) for every x1, x2, y1, and y2, we could demand that:
1. f(x1y1,x2) 6 f(x1,x2)+ f(y1,x2) for every x1, x2, and y1; and
2. f(x1,x2y2) 6 f(x1,x2)+ f(x1,y2) for every x1, x2, and y2. 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 > 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 δ > 0 there
2 exists R> 0 such that, if both x1 > R and x2 > R, then: f(x ,x ) f(x ,x ) 1 2 < inf 1 2 + δ . x x x1,x2>0 x x 1 · 2 1 · 2 In addition, f(x ,x ) f(x ,x ) f(x ,x ) lim lim 1 2 = lim lim 1 2 = inf 1 2 . x1→+∞ x2→+∞ x x x2→+∞ x1→+∞ x x x1,x2>0 x x 1 · 2 1 · 2 1 · 2 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 (x1,x2) 6 (y1,y2) if and only if x1 6 y1 and x2 6 y2, then the f(x ,x ) simultaneous limit on this directed set of the net 1 2 is its greatest x1 x2 lower bound. This is a generalization of the original statem· ent, 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 inte- gers (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 Rd 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] [c, d] for every 0 < a < b and 0 3 2 Background Throughout the paper, the subsets of Rd and the real-valued functions of real variables are presumed to be Lebesgue measurable. We also let real-valued functions take value either + or , but not both. ∞ −∞ We denote by R, R+, and R− the sets of real numbers, positive real numbers, and negative real numbers, respectively. Similarly, we denote by Z, Z+, and Z− the sets of integers, positive integers, and negative integers, respectively. All these sets are considered as additive semigroups (groups in the case of R and Z). If m and n are integers and m 6 n we denote the slice m,m + 1,...,n 1,n = [m,n] Z as [m:n]. If X is a set, we denote by { − } ∩ (X) the set of its finite subsets. PF 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 E(x) the 7→ function that associates to each value x taken by x the value E(x). For example, x 1 is the constant function taking value 1 everywhere. 7→ A directed set is a partially ordered set = (U, 4) with the following U additional property: for every x,y U there exists z U such that x 4 z ∈ ∈ and y 4 z. Every totally ordered set is a directed set, and so is the family of decompositions x = a = x For example, a subsequence xn k> of a sequence xn n> of real numbers { k } 1 { } 1 is a subnet, with V = U = Z+ and φ(k)= nk. If = (U, 4) is a directed set and f : U R is a function, the lower U → limit and the upper limit of f in are the values U lim inf f(x) = sup inf f(y) (2) x→U x∈U y 4 Moreover, if = (V, 6) is a directed set and g : V R is a subnet of f, V → then: lim inf f(x) 6 lim inf g(y) 6 lim sup g(y) 6 lim sup f(x) . (5) x→U y→V y→V x→U If lim infx→U f(x) = limsupx→U f(x), their common value L is called the limit of f in , and we say that f converges to L in . This is equivalent to U U the following: for every ε > 0 there exists xε U such that f(x) L < ε ∈ | − | for every x < xε. In this case, every subnet of f also converges to L. The ordered product of a family (Xi, 6i) i I of ordered sets is the or- { } ∈ dered set (X, 6Π) where X = i∈I Xi and the product ordering 6Π is defined as: Q x 6 y xi 6 yi for every i I . (6) Π ⇐⇒ ∈ If each (Xi, 6i) is a directed set, then so is (X, 6Π). For d > 2 and w d d ∈ 0, 1 the orthant denoted by w is the directed set w = (Xi, 6i) { } R i=1 where (Xi, 6i) = (R , 6) if wi = 0 and (Xi, 6i) = (R , >) if wi = 1. + − Q For example, is the open second quadrant of the Cartesian plane, with R10 (x1,x2) 6 (y1,y2) if and only if x1 > y1 and x2 6 y2. In particular, the Rd d d Rd main orthant of , corresponding to w = 0 , is + = +, 6Π . Note d d R that, if f : R R is a net on and xi,n n> , i [1:d], are sequences + → R+ { } 1 ∈ of positive reals such that limn xi,n = + for every i [1:d], then →∞ ∞ ∈ g(n) = f (x1,n,...,xd,n) is a subnet of f: consequently, if f converges to L R in d , then g(n) converges to L for n . ∈ R+ →∞ 3 Componentwise subadditivity In the literature, subadditivity is most often studied in functions of a sin- gle 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. d Definition 3.1. Let S1,...,Sd be semigroups, let S = i=1 Si, and let f : S R. Given i [1:d], we say that f is subadditive in xi independently → ∈ Q of the other variables if, however given xj Sj for every j [1:d] i , the ∈ ∈ \ { } function xi f(x ,...,xi,...,xd) is subadditive on Si. We say that f is 7→ 1 componentwise subadditive if it is subadditive in each variable independently of the others. 5 Example 3.2. If f : S R and f : S R are both subadditive and 1 1 → 2 2 → nonnegative, then f : S S R defined by f(x ,x )= f (x ) f (x ) is 1 × 2 → 1 2 1 1 · 2 2 componentwise subadditive. If one between f1 and f2 takes negative values, then f might not be componentwise subadditive. For example, f(x ) = x is subadditive on 1 − 1 R+, because it is linear, and f2(x2)= √x2 is also subadditive on R+, because 2 it is nondecreasing and x2 + y2 < √x2 + √y2 for every x2,y2 > 0; but for any fixed x > 0, the function x x √x is not subadditive on R . 1 2 7→ − 1 2 + Example 3.3. (cf. [1, Section 3]) Let d be a positive integer and let A be a finite set with a > 2 elements, considered as a discrete space. The translation d Zd Zd by v Z is the function σv : A A defined by σv(c)(x)= c(x + v) for ∈ → d every x Zd. A d-dimensional subshift on A is a subset X of AZ which is ∈ closed in the product topology and invariant by translation, that is, if c X, d ∈ then σv(c) X for every v Z . Given d positive integers n1,...,nd, an ∈ ∈ d allowed pattern of sides n1,...,nd for X is a function p : i=1 [1:ni] A d → such that there exists c X for which the restriction of c to [1:ni] ∈ Q i=1 coincides with p. Let X (n ,...,nd) be the number of allowed patterns for A 1 Q X of sides n1,...,nd: then f(n ,...,nd) = log X (n ,...,nd) for every n ,...,nd Z (7) 1 a A 1 1 ∈ + is componentwise subadditive, because every allowed pattern of sides n1 + m1,n2,...,nd can be obtained by joining an allowed pattern of sides n1,n2, ...,nd with an allowed pattern of sides m1,n2,...,nd, 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 trans- − lations. Componentwise subadditivity is very different from subadditivity with respect to the operation of the product semigroup. Already with d = 2, if f : S S R is subadditive, then for every x ,y S and x ,y S 1 × 2 → 1 1 ∈ 1 2 2 ∈ 2 we have: f(x1y1,x2y2) 6 f(x1,x2)+ f(y1,y2) , (8) while if f is componentwise subadditive, then for every x ,y S and 1 1 ∈ 1 x ,y S we have the more complex upper bound: 2 2 ∈ 2 f(x1y1,x2y2) 6 f(x1,x2)+ f(x1,y2)+ f(y1,x2)+ f(y1,y2) . (9) 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. 6 Example 3.4. By our discussion in Example 3.2, the function f(x1,x2)= R2 √x1x2 is componentwise subadditive on +. However, f is not subadditive, because f(3, 3) = 3 > 2√2= f(1, 2) + f(2, 1). Example 3.5. The function (7) of Example 3.3 is not, in general, subad- 2 ditive. For example, for d = 2 and X = AZ every pattern is allowed, so f(n1,n2) = n1n2: but if n1, n2, m1, and m2 are all positive, then (n1 + m1)(n2 + m2) >n1n2 + m1m2. Although componentwise subadditivity is very different from subaddi- tivity in the product semigroup, Fekete’s lemma can tell us something im- portant for the case of positive integer or real variables. d R Z Lemma 3.6. Let S = i=1 Si with each Si being either + or +, and let f : Sd R be componentwise subadditive. Having fixed k → ∪ {−∞}Q ∈ [1:d 1], let i, j ,...,jk [1:d] be pairwise different. However fixed the − 1 ∈ values of the remaining variables, the function h : Si R defined → ∪ {−∞} by the multiple limit: f(x1,...,xd) h(xi) = lim . . . lim k (10) xj →+∞ xj →+∞ 1 k i=1 xji is subadditive. Q Proof. It is sufficient to prove the thesis for k = 1; the general case fol- lows by repeated application. To simplify notation, let j = j1. Fix the values of xs for s [1:d] i, j . By hypothesis, for every xi Si the ∈ \ { } ∈ function xj f(x ,...,xd) is subadditive, so by Fekete’s lemma h(xi) = 7→ 1 lim f(x1,...,xd) exists. But for every x ,x′ ,x > 0 it is: xj →∞ xj i i j f(...,x + x′ ,...) f(...,x ,...) f(...,x′ ,...) i i 6 i + 1 , xj xj xj ′ ′ so it must be h(x1 + x1) 6 h(x1)+ h(x1) too. Note that it is crucial for the proof that xj > 0. The following observation is crucial for the next sections. Proposition 3.7. Let w = w1 . . . wd be a binary word of length d and let wi f : Rw R. For every i [1:d] let xw,i = ( 1) xi R+, and let d → ∈ − ∈ fw : R R be defined by fw(xw, ,...,xw,d)= f(x ,...,xd). The following + → 1 1 are equivalent: 7 1. f(x1,...,xd) is componentwise subadditive in Rw. Rd 2. fw(xw,1,...,xw,d) is componentwise subadditive in +. d d d The same holds if Rw and R are replaced with Zw = Rw Z and Z , + ∩ + 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 Zd ; [1, Theorem 1]). Let = Zd , 6 + U + Π and let f : Zd R be componentwise subadditive. Then: + → f(x ,...,x ) f(x ,...,x ) lim 1 d = inf 1 d . (11) Z (x1,...,xd)→U x xd x1,...,xd∈ + x xd 1 · · · 1 · · · Example 4.2. With the notation of Example 3.3 and as in Proposition U 4.1, the value: log X (x ,...,xd) h(X) = lim a A 1 (12) (x1,...,xd)→U x xd 1 · · · 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 [s, 2s), and every y > 0 large enough has a unique ∈ writing y = mt + p with m positive integer and p [t, 2t). By subadditivity, ∈ f(x,y) q 1 6 f(s,y)+ f(r, y) x y x y x y · q · m · 6 f(s,t) x · y · q 1 1 m + f(s,p)+ f(r, t) x · y · x · y · 1 1 + f(r, p) x · y · Consider the four summands on the right-hand side of the last inequality. By construction, limx→+∞ q/x = 1/s and limy→+∞ m/y = 1/t: therefore, the first summand converges to f(s,t)/st for (x,y) 2 . → R+ 8 Now, by [6, Theorem 6.4.1], a subadditive function of one positive real variable is bounded in every compact subset of R . Then p f(s,p) is + 7→ bounded on [t, 2t] and r f(r, t) is bounded on [s, 2s]: consequently, the 7→ second and third summand vanish for (x,y) 2 . → R+ But the fourth summand presents a problem. What we know, is that x f(x,y) is bounded in [s, 2s] for every y [t, 2t], and y f(s,y) is 7→ ∈ 7→ bounded in [t, 2t] for every x [s, 2s]. This is, in general, strictly less than ∈ f being bounded in [s, 2s] [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 : R 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 : R2 R defined by + → f(x,y) = min(h(x), h(y)) satisfies the following conditions: 1. for every x [1, 2], the function y f(x,y) is bounded in [1, 2]; ∈ 7→ 2. for every y [1, 2], the function x f(x,y) is bounded in [1, 2]. ∈ 7→ However, f is not bounded in [1, 2] [1, 2], because f(1 + 1/n, 1 + 1/n)= n × for every n Z+. On the other hand, h(4) = 1 and h(π)= h(4 π) = 0, so ∈ R2 − f is neither subadditive nor componentwise subadditive in +. 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 : Rd R + → and t ,...,td R , let: 1 ∈ + d Vt ,...,t ,k = (x ,...,xd) R 0 (w1) (wd) Vw = y ,...,y (x ,...,xd) V 1 d | 1 ∈ n o is measurable and satisfies µ(Vw)= µ(V ). Note that V = V0d . By repeatedly applying subadditivity, once in each variable, we arrive at: (w1) (wd) f(t1,...,td) 6 f y1 ,...,yd . (15) d w∈{X0,1} For example, for d = 2 we have: f(t ,t ) 6 f(x ,t )+ f(t x ,t ) 1 2 1 2 1 − 1 2 6 f(x ,x )+ f(x ,t x ) 1 2 1 2 − 2 +f(t x ,x )+ f(t x ,t x ) . 1 − 1 2 1 − 1 2 − 2 For (15) to hold, at least one of the 2d summands on the right-hand side f(t1,...,td) d must be no smaller than . Then d V = (0,t ), so: 2d w∈{0,1} w i=1 i S d Q t1 td 6 µ(Vw) = 2 µ(V ) . · · · d w∈{X0,1} From Lemma 4.4 follows: Rd R Theorem 4.5. Let f : + be componentwise subadditive. Then f is → Rd bounded in every compact subset of +. 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 d n > 1 and i [1:d] there exists xi,n [a, b] such that f(x ,n,...,xd,n) > 2 n. ∈ ∈ 1 Let Wt1,...,td be the set in (14). By construction, for every n > 1 we have Wx ,...,x Vb,...,b,n , 1,n d,n ⊆ and by Lemma 4.4, x1,n xd,n a d µ Wx ,...,x > · · · > . 1,n d,n 2d 2 10 Now, the sets Vb,...,b,n are measurable and form a nonincreasing sequence, so d V = n>0 Vb,...,b,n is measurable and µ(V ) > (a/2) : in particular, V cannot be empty. But for (x ,...,xd) V it must be f(x ,...,xd) > n for every T 1 ∈ 1 n > 1: which is impossible. Next, suppose that f is unbounded from below in H. Then for every n > 1 and i [1:d] there exists xi,n [a, b] such that f(x ,n,...,xd,n) 6 n: ∈ ∈ 1 − we may assume that limn→∞ xi,n = xi [a, b] exists for every i [1:d]. Let d∈ ∈ s = min(a, 1), t = b + 4, and J = [s,t] : then every point (z1,...,zd) where each zi belongs to either [a, b] or [1, 4] belongs to J. Let now yi [1, 4] for ∈ every i [1:d] and ∈ M = sup f(z ,...,zd) (z ,...,zd) J , { 1 | 1 ∈ } which is a real number because of the previous point. By applying subad- ditivity in each variable, for such y1,...,yd and n we obtain d f(y + x ,n,...,yd + xd,n) 6 (2 1)M n , 1 1 − − because n is an upper bound for f(x ,n,...,xd,n) and M is an upper bound − 1 for the other 2d 1 summands, For example, for d = 2 we have: − f(y1 + x1,n,y2 + x2,n) 6 f(y1,y2)+ f(y1,x2,n) +f(x1,n,y2)+ f(x1,n,x2,n) 6 3M n . − But for every n such that xi,n xi 6 1 it is [xi +2,xi +3] [xi,n +1,xi,n +4]: | − | ⊆ calling d K = [xi + 2,xi + 3] J , ⊆ i Y=1 for every n large enough every element of K can be written in the form (y1 + x1,n,...,yd + xd,n) for suitable y1,...,yd [1, 4]. For every (z1,...,zd) K d ∈ ∈ it must then be f(z ,...,zd) 6 (2 1)M n for every n large enough: 1 − − which is impossible. If f : Rw R is bounded on the compact subsets of Rw, then fw as → defined in Proposition 3.7 is bounded on the compact subsets of R+; and vice versa. From Theorem 4.5 and Proposition 3.7 follows: d Corollary 4.6. Let w 0, 1 and let f : Rw R be componentwise ∈ { } → subadditive. Then f is bounded in every compact subset of Rw. 11 In turn, Corollary 4.6 allows us to prove: Theorem 4.7. Let f : Rd R be componentwise subadditive. Then f is → bounded in every compact subset of Rd. Proof. It is sufficient to show finitely many open sets V1,...,Vn such that f is bounded on the compacts of each Vi and: n d R = Rw Vi . ∪ d i ! w∈{[0,1} [=1 We give the argument for d = 3: the ideas for arbitrary d > 1 are similar. Let I = [ 1/2, 1/2] and U = [ 3/2, 1/2] [1/2, 3/2]. − − − ∪ We start by proving that f is bounded in every compact subset of the open set Z = (x,y,z) R3 x> 0,y > 0 = R R D , 00 { ∈ | } 000 ∪ 001 ∪ 00 where D = (x,y,z) R3 x > 0,y > 0, z = 0 is the first quadrant 00 { ∈ | } 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] [a, b] I. Let V = [a, b] [a, b] U: if × × × × (x,y,z) 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) H, ∈ f(x,y,z) 6 f(x,y,z 1) + f(x,y, 1) 6 2T − and f(x,y,z) > f(x,y,z + 1) f(x,y, 1) > t T . − − By similar arguments, f is bounded in every compact subset of every subset of R3 which is the union of two adjacent orthants and the corresponding “quadrant”. As for each open orthant there are three which border it by 8 3 one “quadrant”, there are · = 12 such subsets. 2 We now show that f is bounded in every compact subset of the open “upper demispace” Z = (x,y,z) R3 z > 0 . To do so, it will suffice 0 { ∈ | } to show that f is bounded in every set of the form L = I I [a, b] with × × 0 < a < b. Let W = U U [a, b] and let S and s be an upper bound for × × f in W , respectively: then for every (x,y,z) L, ∈ f(x,y,z) 6 f(x 1,y,z)+ f(1,y,z) − 6 f(x 1,y 1, z)+ f(x 1, 1, z)+ f(1,y 1, z)+ f(1, 1, z) − − − − 6 4S 12 and f(x,y,z) > f(x + 1,y,z) f(1,y,z) − > f(x + 1,y + 1, z) f(x + 1, 1, z) f(1,y,z) − − > s 2S . − 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 I I. Let E = U U 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) K, ∈ f(x,y,z) 6 f(x 1,y,z)+ f(1,y,z) − 6 f(x 1,y 1, z)+ f(x 1, 1, z)+ f(1,y 1, z)+ f(1, 1, z) − − − − 6 f(x 1,y 1, z 1) + f(x 1, z 1, 1) − − − − − +f(x 1, 1, z 1) + f(x 1, 1, 1) − − − +f(1,y 1, z 1) + f(1,y 1, 1) − − − +f(1, 1, z 1) + f(1, 1, 1) − 6 8M and f(x,y,z) > f(x + 1,y,z) f(1,y,z) − > f(x + 1,y + 1, z) f(x + 1, 1, z) f(1,y,z) − − > f(x + 1,y + 1, z + 1) f(x + 1,y + 1, 1) − f(x + 1, 1, z) f(1,y,z) − − > m 3M . − Note that the argument of Lemma 4.4 also works if f is subadditive, rather than componentwise subadditive. In this case, however, the denomi- nator in (14) and in the thesis is 2 rather than 2d. A more complex variant of it can then be stated, where f is a function of k variables xi, each taking Rdi k values in an orthant of + : and the denominator would then be 2 . From this, a generalization of Theorem 4.5 to the case of componentwise functions Rdi of k variables, the ith of which takes values in + , can be derived. 13 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 Rd ). Let d > 1 and let f : Rd R be + + → componentwise subadditive. Then: f(x ,...,x ) f(x ,...,x ) lim 1 d = inf 1 d , (16) d R (x ,...,x )→R x xd x1,...,xd∈ + x xd 1 d + 1 · · · 1 · · · which can be . In addition, for every permutation σ of [1:d], −∞ f(x ,...,x ) f(x ,...,x ) lim . . . lim 1 d = lim 1 d , (17) xσ(1)→+∞ xσ(d)→+∞ x xd (x ,...,x )→Rd x xd 1 · · · 1 d + 1 · · · 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 > 1 variables xi, no matter of what type. Then for every permutation σ of [1:d], inf . . . inf u(x1,...,xd)= inf u(x1,...,xd) . (18) xσ(1) xσ(d) x1,...,xd Sketch of proof. Let L and R be the left- and right-hand side of (18), re- spectively. It is easy to see that L > R. Let now δ > 0: there exist z ,...,zd R such that u(z ,...,zd) < R + δ, so a fortiori L < R + δ too; 1 ∈ + 1 as δ > 0 is arbitrary, L 6 R. Proof of Theorem 5.1. For every i 1,...,d and x ,...,xd R , if xi = ∈ { } 1 ∈ + qt + r with q Z and r, t R , then: ∈ + ∈ + f(x ,...,xi,...,xd) 6 q f(x ,...,t,...,xd)+ f(x ,...,r,...,xd) . 1 · 1 1 Fix t ,...,td R . For every i [1:d] and xi > 2ti there exist unique 1 ∈ + ∈ qi Z+ and ri [ti, 2ti) such that xi = qiti + ri. For every i [1:d] let (0)∈ (1)∈ ∈ yi = ri and yi = ti: by repeatedly applying subadditivity, once per each variable, we find: w1 wd (w1) (wd) f(x1,...,xd) 6 q1 qd f y1 ,...,yd . (19) d · · · · w∈{X0,1} 14 Now, on the right-hand side of (19), each occurrence of f has k arguments chosen from the ti’s and d k chosen from the ri’s, is multiplied by the qi’s − corresponding to the ti’s, and is bounded from above by the constant M = sup f(y ,...,yd) yi [ti, 2ti] i [1:d] , { 1 | ∈ ∀ ∈ } 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 xd we get: 1 · · · w1 wd f(x1,...,xd) q1 qd q1 qd 6 · · · f(t1,...,td)+ M · · · . (20) x1 xd x1 xd · d d x1 xd · · · · · · w∈{0,X1} \{1 } · · · where 1d is the binary word of length d where all the bits are 1. (ε) (ε) By construction, limxi→∞ qi/xi = 1/ti. Given ε> 0, choose x1 ,...,xd (ε) ∈ R such that, if xi > x for each i [1:d], then the following hold: + i ∈ f(x1,...,xd) f(t1,...,td) ε 1. < + d ; x xd t td 2 1 · · · 1 · · · w1 wd q1 qd ε d d 2. · · · < d for every w 0, 1 1 . x xd M 2 ∈ { } \ { } 1 · · · · This is possible because if w = 1d, then at least one of the qwi equals 1. For 6 i such x1,...,xd it is: f(x ,...,x ) f(t ,...,t ) 1 d < 1 d + ε . x xd t td 1 · · · 1 · · · As ε> 0 is arbitrary, it must be: f(x ,...,x ) f(t ,...,t ) lim sup 1 d 6 1 d . (x ,...,x )→Rd x1 xd t1 td 1 d + · · · · · · But the ti’s are also arbitrary, hence: f(x ,...,x ) f(t ,...,t ) lim sup 1 d 6 inf 1 d R (x ,...,x )→Rd x1 xd t1,...,td∈ + t1 td 1 d + · · · · · · f(x ,...,x ) 6 lim inf 1 d , (x ,...,x )→Rd x xd 1 d + 1 · · · which yields (16). 15 Now, by Lemma 3.6, for every choice of i, j ,...,jk [1:d] all different, 1 ∈ 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: f(x ,...,x ) f(x ,...,x ) lim . . . lim 1 d = inf lim . . . lim 1 d xσ(1)→+∞ xσ(d)→+∞ x1 xd xσ(1)>0 xσ(2)→+∞ xσ(d)→∞ x1 xd · · · = . . . · · · f(x ,...,x ) = inf . . . inf 1 d xσ(1)>0 xσ(d)>0 x xd 1 · · · f(x ,...,x ) = inf 1 d x1,...,xd>0 x xd 1 · · · f(x ,...,x ) = lim 1 d . (x ,...,x )→Rd x xd 1 d + 1 · · · From Theorem 5.1 and Proposition 3.7 follows: ′ d Theorem 5.3. Let d > 1, let w, w 0, 1 and let f : Rw R be ∈ { } → componentwise subadditive. 1. If w contains evenly many 1s, then: f(x ,...,x ) f(x ,...,x ) lim 1 d = inf 1 d (21) (x1,...,xd)→Rw x xd (x1,...,xd)∈Rw x xd 1 · · · 1 · · · is not + , but can be . ∞ −∞ 2. If w contains oddly many 1s, then: f(x ,...,x ) f(x ,...,x ) lim 1 d = sup 1 d (22) (x1,...,xd)→Rw x xd R x xd 1 · · · (x1,...,xd)∈ w 1 · · · is not , but can be + . −∞ ∞ 3. Suppose now w contains evenly many 1s, w′ differs from w in exactly one coordinate, and f is defined and componentwise subadditive in Rw Rw′ Uw,w′ , where: ∪ ∪ d Uw,w′ = x R xi = 0, (x ,...,xi , 1,xi ,...,xd) Rw Rw′ { ∈ | 1 −1 +1 ∈ ∪ } is the boundary between Rw and Rw′ . Then: f(x ,...,x ) f(x ,...,x ) lim 1 d 6 lim 1 d ; (23) (x1,...,xd)→R ′ x xd (x1,...,xd)→Rw x xd w 1 · · · 1 · · · consequently, both limits are finite. 16 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 R be a subadditive function. → If S is a monoid with identity e, then f(e) > 0. If, in addition, S is a group, then f(x)+ f(x−1) > 0 for every x S. ∈ Proof of Theorem 5.3. For (x ,...,xd) w and i [1:d] let xw,i and 1 ∈ R ∈ fw be defined as in Proposition 3.7. If w contains evenly many 1s, then x xd = xw, xw,d and: 1 · · · 1 · · · f(x ,...,x ) fw(xw, ,...,x ) lim 1 d = lim 1 w,d d x→Rw x xd xw→R xw, xw,d 1 · · · + 1 · · · fw(xw, ,...,x ) = inf 1 w,d x ∈Rd xw, xw,d w + 1 · · · f(x ,...,x ) = inf 1 d . x∈Rw x xd 1 · · · If w contains oddly many 1s, then x xd = xw, xw,d and: 1 · · · − 1 · · · f(x ,...,x ) fw(xw, ,...,x ) lim 1 d = lim 1 w,d x→Rw x xd − x →Rd xw, xw,d 1 · · · w + 1 · · · fw(xw, ,...,x ) = inf 1 w,d − x ∈Rd xw, xw,d w + 1 · · · f(x ,...,x ) = sup 1 d . x R x xd ∈ w 1 · · · Suppose now w has evenly many 1s and w′ differs from w only in component i, and f is defined and componentwise subadditive in Rw Rw′ Uw,w′ . Then ∪ ∪ for every x ,...,xi ,xi ,...xd the function xi f(x ,...,xi,...,xd) is 1 −1 +1 7→ 1 subadditive on R: by Lemma 5.4, for every xi R, ∈ f(x ,...,xi,...,xd)+ f(x ,..., xi,...,xd) > 0 . 1 1 − Then: ′ ′ f(x1,...,xd) f(x1,...,xd) lim ′ lim ′ ′ x→Rw x xd − x →Rw′ x x 1 · · · 1 · · · d f(x ,...,xi,...,xd) f(x ,..., xi,...,xd) = lim 1 + lim 1 − x→Rw x xd x→Rw x xd 1 · · · 1 · · · f(x ,...,xi,...,xd)+ f(x ,..., xi,...,xd) = lim 1 1 − x→Rw x xd 1 · · · 17 is nonnegative. The last passage is valid because the two limits on the second line are either finite or . −∞ As every subnet of a convergent net converges to the same limit, we get: Corollary 5.5. Let f : Rd R be componentwise subadditive and let + → T be either R or Z . For every i [1:d] let xi(t) : T R satisfy + + ∈ → + limt xi(t)=+ . Then: →+∞ ∞ f(x (t),...,x (t)) f(x ,...,x ) lim 1 d = inf 1 d , (24) t→+∞ x (t) xd(t) x1,...,xd>0 x xd 1 · · · 1 · · · and also f(x (t),...,x (t)) f(x ,...,x ) inf 1 d = inf 1 d . (25) t∈T x (t) xd(t) x1,...,xd>0 x xd 1 · · · 1 · · · In particular, f(n,...,n) f(n,...,n) f(x1,...,xd) lim d = inf d = inf . (26) n→∞ n n>1 n x1,...,xd>0 x xd 1 · · · Sketch of proof. We only remark that (25) follows from (16) and: f(x ,...,x ) f(x (t),...,x (t)) inf 1 d 6 inf 1 d x1,...,xd>0 x xd t∈T x (t) xd(t) 1 · · · 1 · · · f(x (t),...,x (t)) 6 lim inf 1 d . t→∞ x (t) xd(t) 1 · · · Rd Note that, in general, even if f is componentwise subadditive on +, t f(t,...,t) is not subadditive on R : a simple example is f(x ,x ) = 7→ + 1 2 x x . This provides further evidence that Theorems 5.1 and 5.3 are not 1 · 2 special cases of [6, Theorem 6.6.1]. A real-valued function defined on a semigroup (S, ) is superadditive if it · satisfies f(x y) > f(x)+ f(y) for every x,y S. As f is superadditive if · ∈ and only if f is subadditive, an analogue of Theorem 5.1 holds for com- − ponentwise superadditive functions, provided one swaps the roles of inf and sup and those of and + . 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 : R2 R defined by f(x ,x )= x2√x is su- + → 1 2 1 2 peradditive in x1 and subadditive in x2, and f(x1,x2)/x1x2 = x1/√x2. But lim 2 f(x1,x2)/x1x2 does not exist, because for every y,R > 0 there (x1,x2)→R+ 18 f(x1,x2) exist x ,x > R such that x /√x = y. Also, limx limx = 1 2 1 2 1→∞ 2→∞ x x 1 · 2 f(x1,x2) 0 but limx limx =+ . 2→∞ 1→∞ x x ∞ 1 · 2 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 : Rd R be subadditive → in the variable x Rd. Then for every x Rd the following limit exists: ∈ ∈ f(tx) Lx = lim . t→+∞ t This, however, is not so much an extension than a corollary. If f : Rd → R satisfies f(x + y) 6 f(x)+ f(y) for every x, y Rd, then obviously ∈ gx(t) = f(tx) satisfies gx(s + t) 6 gx(s)+ gx(t) for every s,t > 0: and Lx is simply the limit of gx(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 = (U, 4) and a net U Fx x U of finite nonempty subsets of G such that: { } ∈ gFx Fx lim | \ | = 0 for every g G . (27) x→U Fx ∈ | | 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 : (G) R be a function which: PF → 1. is subadditive with respect to set union, that is, f(A B) 6 f(A)+f(B) ∪ for every A, B (G); and ∈ PF 2. satisfies f(A)= f(gA) for every A (G) and g G. ∈ PF ∈ Then for every directed set = (U, 4) and every left Følner net = U F Fx x∈U on G, { } f(F ) L = lim x (28) x→U Fx | | exists, and does not depend on the choice of and . U F 19 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 G G a > 2 elements. The shift by g G is the function σg : A A defined by ∈ G → σg(c)(x)= c(g x) for every c A and x G. The notions of subshift and · ∈ ∈ of allowed pattern with support S (G) are extended naturally from ∈ PF those of Example 3.3. Calling X (S) the number of allowed patterns for A X with support S, and convening that the unique empty pattern e : A ∅→ appears in every configuration, we have for every S, T (G): ∈ PF X(S) 6 X (S T ) 6 X (S) X (T S) 6 X (S) X (T ) . A A ∪ A ·A \ A ·A Indeed, every allowed pattern on S (resp., T S) can be extended to at least \ one allowed pattern on S T (resp., T ) but joining an allowed pattern over ∪ S and an allowed pattern over T S does not necessarily yield an allowed \ pattern on S T . Hence, f(S) = log X (S) is subadditive on (G), and ∪ a A PF clearly satisfies f(gS)= f(S) for every g G and S (G). The entropy ∈ ∈ PF of X can then be defined as: log X (Fx) h(X) = lim a A (29) x→U Fx | | where = (U, 4) is an arbitrary directed set and Fx x U is an arbitrary U { } ∈ Følner net on G. d As the sets Ex ,...,x = [1:xi] with x ,...,xd Z constitute a 1 d i=1 1 ∈ + Følner net on Zd, defining the entropy of a d-dimensional subshift according Q 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 : Z → R is subadditive, the “natural” conversion f( A ) if A = , g(A)= | | 6 ∅ (30) 0 otherwise , where A is the number of elements of A, is invariant by translations, but | | needs not be subadditive on (Z), the main reason being that A B PF | ∪ | 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. 20 Example 6.3. The function f(n)= n mod 2 is easily seen to be subadditive on Z. But the function g defined from f by (30) is not subadditive on (Z), because if U = 1, 2 and V = 2, 3 , then g(U V ) = 1 and PF { } { } ∪ 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 > 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. References [1] Silvio Capobianco. Multidimensional cellular automata and generaliza- tion of Fekete’s lemma. Discrete Mathematics and Theoretical Com- puter Science, 10:95–104, 2008. dmtcs.episciences.org/442. [2] Tullio G. Ceccherini-Silberstein and Michel Coornaert. Cellular Au- tomata and Groups. Springer Verlag, 2010. [3] Mih´aly Fekete. Uber¨ die Verteilung der Wurzeln bei gewissen alge- braischen Gleichungen mit ganzzahlingen Koeffizienten. Mathematische Zeitschrift, 17:228–249, 1923. doi:10.1007/BF01504345. [4] Erling Følner. On groups with full Banach mean value. Mathematica Scandinavica, 3:245–1254, 1955. doi:10.1007/s00010-012-0178-2. [5] Francesco Guerra and Fabio Lucio Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics, 230:71–79, 2002. doi:10.1007/s00220-002-0699-y. [6] Einar Hille. Functional Analysis and Semi-groups. American Mathe- matical Society, 1948. 21 [7] Fabrice Krieger. The Ornstein-Weiss lemma for discrete amenable groups. Technical report, Max Planck Institute for Mathe- matics Bonn, 2010. MPIM Preprint 2010–48. Available from G¨ottingen State and University Library. [8] Marek Kuczma. An Introduction to the Theory of Functional Equations and Inequalities. Springer, 2009. [9] Douglas Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1995. [10] Erez Louidor, Brian Marcus, and Ronnie Pavlov. Independence entropy of Zd shift spaces. Acta Applicandae Mathematicae, 126:297–317, 2013. doi:10.1007/s10440-013-9819-2. [11] Donald S. Ornstein and Benjamin Weiss. Entropy and isomorphism the- orems for actions of amenable groups. Journal d’Analyse Math´ematique, 48:1–141, 1987. doi:10.1007/BF02790325. [12] Wac law Sierpi´nski. Sur l’´equation fonctionnelle f(x + y) = f(x) + f(y). Fundamenta Mathematicae, 1:116–122, 1920. Available from the Polish Digital Mathematics Library. 22