Annales Mathématiques Blaise Pascal

Georges Grekos

Vladimír Toma

Jana Tomanová

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Georges Grekos, Vladimír Toma and Jana Tomanová
A note on uniform or Banach density
Volume 17, no 1 (2010), p. 153-163.
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Annales mathématiques Blaise Pascal 17, 153-163 (2010)
A note on uniform or Banach density
Georges Grekos
Vladimír Toma
Jana Tomanová
Abstract
In this note we present and comment three equivalent definitions of the so
called uniform or Banach density of a set of positive integers.
Introduction
The concept of density of an infinite set of positive integers appeared to
be a basic tool to obtain an idea of the magnitude and of the structure
of subsets of positive integers. During the last decades several types of
densities have been introduced and studied. In light of this, both ques-
tions on relationships between various kinds of densities and various, but
equivalent, ways to define a given type of density are interesting.
The most frequently used density concept is that of asymptotic (or
natural) density. The upper (resp. lower) asymptotic density of a set A of
positive integers, denoted by d(A) (resp. d(A)), is the upper (lower) limit,
as n tends to infinity, of the “local density” of A in the initial interval
[1, n], that is of
1
n
|A ∩ [1, n] |.
N.B.- If M is a finite subset of N, then the symbol |M | stands for the
cardinality of M .
The aim of this note is to present and comment three equivalent def-
initions of the so called uniform or Banach density of a set of positive
integers. Given a set A ⊆ N, its upper and lower Banach densities, de-
noted by b(A) and b(A) (see definitions in sections 1 and 3, respectively),
satisfy the relation
0 ≤ b(A) ≤ d(A) ≤ d(A) ≤ b(A) ≤ 1. (0.1)
Keywords: Banach density, uniform density.
Math. classification: 11B05.
153
G. Grekos, V. Toma and J. Tomanová
It is easy to prove that these densities can have any prescribed values,
belonging to the interval [0,1] and satisfying the inequalities in relation
(0.1).
The principal characteristic of the uniform density is that it is more
sensitive to local density in any interval, not necessarily initial, than the
asymptotic density. For instance the set
A = ∪n≥1{n! + 1, n! + 2, . . . , n! + n},
having rare but sufficiently long blocks of consecutive integers, has asymp-
totic density zero while its upper uniform density equals to 1 and so its
uniform density does not exist.
In what follows N denotes the set of all non negative integers and N∗ the
set of all positive integers. If s, t are integers, 0 ≤ s ≤ t, then I = [s, t]∩N
denotes an interval in N.
1. The definitions and our setting
1.1. Banach density
In [1], [2], [11] and [12], for instance, the notion of Banach density is used.
If A is a subset of N, then we put
E = E(A) = {x ∈ [0, 1];∀l ∈ N∗∃I ⊆ N∗ : |I| ≥ l ∧ |A ∩ I|
|I|
≥ x}.
Obviously, the set E is a subinterval of [0, 1] containing 0. The number
b(A) = supE is called the (upper) Banach density of the set A. See
also [13], [7, page 235] and [9, page 72] for a slightly different formulation
of the above definition. For simplicity, we shall write b instead of b(A) in
sections 1 and 2.
1.2. Uniform density
In the papers [3], [4] and [5], the following measure of magnitude of a set
A ⊆ N, based on translating intervals [k, k + h] ∩N of N, was introduced.
If k, h are integers, 0 ≤ k, 0 ≤ h, then we put
A(k, k + h) = |A ∩ [k, k + h]|.
154
A note on uniform or Banach density
For an integer s, s ≥ 1, we set
αs = lim sup
n→∞
A(n+ 1, n+ s), γs = sup
n≥0
A(n+ 1, n+ s).
The numbers αs, γs are integers from the set {0, 1, . . . , s}, satisfying the
inequality αs ≤ γs. The upper uniform density of A is defined either
as (see [4])
a = lim
s→∞
αs
s
or as (see [3], [5])
c = lim
s→∞
γs
s
.
The existence of these limits is often stated without proof or any reference
to a proof (see [4], [5], [16]).
In [15] and [17] it is proved that the limit a exists. Proofs in those papers
are of similar nature, each one using an euclidean division. P. Ribenboim
[15] observes that the function s 7→ αs is subadditive and then proves the
existence of the limit a. The existence of each of the limits a and c (but
not their equality) follows directly from the subadditivity of each of the
functions s 7→ αs and s 7→ γs by the following lemma, attributed to Fekete
[8] (see also [14, 18, 6]).
Lemma 1.1. If the real sequence (an)n≥1 satisfies
am+n ≤ am + an
for all m and n, then limn→∞ ann exists and is equal to infn∈N∗
an
n . This
value is either a real number or −∞.
In the paper [10] the equality a = c is stated without proof. As it was
said in section 1.1, some authors prefer the Banach density b. The advan-
tage is that no proof of existence is needed. Of course in the definitions
of a and c one can put lim sup (or lim inf) instead of lim . But for ap-
plications it is convenient to know that the limit exists. We haven’t seen
anywhere the statements a = b and b = c. However, the last one is implicit
in Bergelson’ papers [1, 2, 3]).
155
G. Grekos, V. Toma and J. Tomanová
1.3. Our contribution
We present in this note a single proof of the existence of a, of the existence
of c and of the equalities
a = c = b = inf
s∈N∗
αs
s
= inf
s∈N∗
γs
s
.
Our proof does not use Lemma 1.1. It is partly inspired from a draft
by the late Professor Tibor Šalát where he proved the existence of a.
Here is the approach. Let us introduce the following upper and lower
limits :
a = lim sup
s→∞
αs
s
, a = lim inf
s→∞
αs
s
,
c = lim sup
s→∞
γs
s
, c = lim inf
s→∞
γs
s
.
Clearly, the numbers defined above are related by
0 ≤ a ≤ min{a, c} ≤ max{a, c} ≤ c ≤ 1.
Further let
a = inf
s∈N∗
αs
s
, c = inf
s∈N∗
γs
s
.
Obviously we have
0 ≤ a ≤ min{a, c} ≤ max{a, c} ≤ c ≤ c ≤ 1.
The main idea in our proof is to compare the above introduced quanti-
ties to b. We shall prove in the next section the following proposition.
Proposition 1.2. We have b ≤ a and c ≤ b.
Proposition 1.2 straightforwardly implies the following theorem.
Theorem 1.3. We have a = a and c = c, so both limits a and c exist
and moreover a = b = c = a = c.
Remark 1.4. The limit a is usually denoted by u (or u(A)) and called the
upper uniform density (of the set A) (see [5]). In the literature we have
examined, the upper Banach density is also denoted by BD(A) (see [12]).
156
A note on uniform or Banach density
2. Proof of Proposition 1.2
2.1. Demonstration of c ≤ b
The case c = 0 is trivial. Suppose c > 0. It suffices to show that [0, c) ⊆ E,
i.e. that for each x ∈ [0, 1], the following implication holds
0 ≤ x < c =⇒ x ∈ E.
Since zero belongs to E we may assume that 0 < x < c. By the definition
of c, there exists a sequence (s1 < s2 < . . . ) of integers such that for every
i ∈ N∗ we have
γsi
si
> x,
i.e.
γsi = sup
n≥0
A(n+ 1, n+ si) > xsi.
The supremum being effectively a maximum, there exists ni ∈ N satisfying
A(ni + 1, ni + si) > xsi.
The last inequality implies that x belongs to E. 2
2.2. Demonstration of b ≤ a
We need to prove that
b ≤ inf
s≥1
αs
s
.
In other words, that for any s ≥ 1, we have
b ≤ α
s
s
.
We proceed indirectly. Suppose that for some integer s0 ≥ 1, we have
αs0
s0
< b.
Fix two reals x1, x2 ∈ [0, 1] such that
αs0
s0
< x1 < x2 < b.
We shall show that x2 does not belong to E, which gives a contradiction.
157
G. Grekos, V. Toma and J. Tomanová
There exists a multiple of s0, say n0, such that the following implication
is true
n ≥ n0 =⇒ A(n+ 1, n+ s0) < x1s0. (2.1)
In order to get the conclusion x2 6∈ E, it suffices to find an l0 ∈ N∗ such
that for each interval I ⊆ N∗ the following implication holds
|I| ≥ l0 =⇒ |A ∩ I| < x2|I|.
To this end we shall consider two cases depending on the form of the
interval I ⊆ N∗. First, let I = [n+ 1, n+ h], where n ≥ n0, h ≥ 1. Taking
into account that s0 is a positive integer, h can be written in the form
h = qs0 + r, with 0 ≤ r < s0.
Consequently
qs0 ≤ h < (q + 1)s0.
Owing to (2.1) and to the above inequality we get
|A ∩ I| ≤ A(n+ 1, n+ s0) +A(n+ s0 + 1, n+ 2s0) + · · ·+
A(n+ qs0 + 1, n+ h) ≤
q+1∑
k=1
A(n+ (k − 1)s0 + 1, n+ ks0)
≤ (q + 1)x1s0.
Second, if n < n0 then the interval I = [n + 1, n + h] can be partitioned
in at most n0+s0s0 subintervals with the same length s0, each of which fails
the upper bound in (2.1). Hence, in general we have
|A ∩ I| ≤ (q + 1)x1s0 + n0 + s0. (2.2)
The right-hand side of (2.2) can be rewritten in the form
qx1s0 + x1s0 + s0 + n0 = hx1 − rx1 + x1s0 + n0 + s0,
therefore
|A ∩ I| ≤ hx1 + x1s0 + n0 + s0 < hx2,
under the condition
h >
x1s0 + n0 + s0
x2 − x1
.
Putting l0 = bx1s0+n0+s0x2−x1 c+ 1 we obtain
|I| = h ≥ l0 =⇒ |A ∩ I| < x2|I|,
which finishes the proof. 2
158
A note on uniform or Banach density
3. Lower Banach density
Similar “dual” results to those we derived in the previous section are valid
for lower density. The property dual to Theorem 1.3 results from the fact
that the lower density of a set A ⊆ N is equal to 1 minus the upper density
of the set Ã = N \ A. This is rather obvious for the uniform density (see
Propositions 3.1 and 3.2 below) and it needs a short proof for the Banach
density (see Proposition 3.3).
The concept of lower uniform density u can be defined for any set A ⊆ N
using the numbers
αs = lim inf
n→∞
A(n+ 1, n+ s), γs = inf
n≥0
A(n+ 1, n+ s).
Obviously, γs ≤ αs and these are integers belonging to {0, 1, . . . , s}.
Proposition 3.1. The limits lims→∞ αss , lims→∞
γs
s exist and are equal.
Proof . We have
Ã(n+ 1, n+ s) = s−A(n+ 1, n+ s).
If we denote by α̃s, γ̃s the values defined in section 1.2, corresponding to
the set Ã, then we get
α̃s
s
= 1− αs
s
,
γ̃s
s
= 1− γs
s
.
This gives
lim
s→∞
αs
s
= 1− lim
s→∞
α̃s
s
= 1− u(Ã),
lim
s→∞
γs
s
= 1− lim
s→∞
γ̃s
s
= 1− u(Ã).
This completes the proof. 2
The common value lims→∞ αss = lims→∞
γs
s =: u(A) is called the lower
uniform density of the set A.
In a similar way one can verify the following property.
Proposition 3.2. We have
u(A) = sup
s≥1
αs
s
= sup
s≥1
γs
s
.
159
G. Grekos, V. Toma and J. Tomanová
Let us turn now to the Banach density. To define the lower Banach
density b(A) of the set A, we introduce the set
F = F (A) = {y ∈ [0, 1];∀l ∈ N∗∃I ⊆ N∗ : |I| ≥ l ∧ |A ∩ I| ≤ y|I|}.
Obviously, the set F is a subinterval of [0, 1] containing 1.
Define the lower Banach density by
b = b(A) = inf F.
The following proposition together with Proposition 3.1 and Theorem 1.3
show that the above definition of lower Banach density gives the equality
b(A) = u(A), as well.
Proposition 3.3. For every A ⊆ N we have b(A) = 1−b(N\A) = 1−b(Ã).
Proof . Let A ⊆ N. For an interval I in N we set
ν(I) = |A ∩ I|
|I|
, ν̃(I) = |Ã ∩ I|
|I|
.
We have
ν̃(I) = 1− ν(I).
So the condition ν(I) ≤ y in the definition of the set F is equivalent to
1− ν̃(I) ≤ y, that is ν̃(I) ≥ 1− y. Recall that
b(Ã) = sup Ẽ,
where
Ẽ = {x ∈ [0, 1];∀l ∈ N∗∃I ⊆ N∗ : |I| ≥ l ∧ ν̃(I) ≥ x}.
The above remark gives
y ∈ F =⇒ 1− y ∈ Ẽ.
Similarly
x ∈ Ẽ =⇒ 1− x ∈ F.
We conclude that
inf F = 1− sup Ẽ
which is the required assertion. 2
Corollary 3.4. For every A ⊆ N we have b(A) = u(A).
160
A note on uniform or Banach density
Once the upper and lower densities are defined, the notion of density
can be introduced as usual: A set A has Banach density provided that
b(A) = b(A), this common value is denoted by b(A). We can conclude
from (0.1) that whenever the Banach density of a set A exists it is equal
to its asymptotic density d(A). However, the concept of Banach density
is interesting even if only the upper and lower Banach densities exist. It
brings more precise information about the spacing of the elements in the
set and also about the extremal distribution of its elements in intervals of
arbitrary length.
Remark 3.5. The term “uniform density” is redundant in the light of
proved results. Since the notion of Banach density has been introduced
prior to the term “uniform density”, we propose to use the term “Banach
density”.
Acknowledgements. We would like to thank the referee for his sug-
gestions and constructive comments. The second and the third author are
grateful to the Jean Monnet University in Saint-Etienne for hospitality
and financial support during their visit that led to the completion of this
work.
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162
A note on uniform or Banach density
Georges Grekos
Department of Mathematics
Université de Saint-Etienne
23, rue du Docteur Paul Michelon
F-42023 Saint-Etienne Cédex 2
France
grekos@univ-st-etienne.fr
Vladimír Toma
Department of Mathematical and
Numerical Analysis
Comenius University
Mlynská dolina
842 48 Bratislava
Slovakia
toma@fmph.uniba.sk
Jana Tomanová
Department of Algebra and
Number Theory
Comenius University
Mlynská dolina
842 48 Bratislava
Slovakia
tomanova@fmph.uniba.sk
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