Comptes Rendus Mathématique

Ji-Zhen Xu

Yong-Gao Chen

Comptes Rendus
Mathématique
Ji-Zhen Xu and Yong-Gao Chen
On subsets of asymptotic bases
Volume 362 (2024), p. 45-49
Online since: 2 February 2024
https://doi.org/10.5802/crmath.513
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 45-49
https://doi.org/10.5802/crmath.513
Research article / Article de recherche
Number theory / Théorie des nombres
On subsets of asymptotic bases
Ji-Zhen Xu a,b and Yong-Gao Chen ∗,a
a School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing
210023, People’s Republic of China
b Nanjing Vocational College of Information Technology,Nanjing 210023, People’s Republic of China
E-mails: 965165607@qq.com (J.-Z. Xu), ygchen@njnu.edu.cn (Y.-G. Chen)
Abstract. Let h ≥ 2 be an integer. In this paper, we prove that if A is an asymptotic basis of order h and B is a
nonempty subset of A, then either there exists a finite subset F of A such that F ∪B is an asymptotic basis of
order h, or for any ε> 0, there exists a finite subset Fε of A such that dL (h(Fε∪B)) ≥ hdL (B)−ε, where dL (X )
denotes the lower asymptotic density of X and hX denotes the set of all x1 +·· ·+ xh with xi ∈ X (1 ≤ i ≤ h).
This generalizes a result of Nathanson and Sárközy.
2020 Mathematics Subject Classification. 11B13, 11B05, 11P99.
Funding. This work is supported by the National Natural Science Foundation of China, Grant No. 12171243.
Manuscript received 25 January 2023, accepted 13 May 2023.
1. Introduction
LetN0 denote the set of all nonnegative integers. Let h ≥ 2 be an integer. For A ⊆N0, let
h A = {a1 +·· ·+ah : a1, . . . , ah ∈ A}.
We define
dL(A) = liminf
x→+∞
A(x)
x
,
where A(x) is the number of positive integers in A which do not exceed x. Usually, dL(A) is called
the lower asymptotic density of A. If
lim
x→+∞
A(x)
x
exists, then the limit value is called the asymptotic density of A and denote it by d(A).
A set A is called an asymptotic basis of order h if h A contains all sufficiently large integers. An
asymptotic basis A of order h is called minimal if no proper subset of A is an asymptotic basis of
order h. The notation of minimal asymptotic bases was introduced by Stöhr [10] in 1955. In 1956,
Härtter [4] proved that for each integer h ≥ 2, there exist minimal asymptotic bases of order h. In
1988, Erdős and Nathanson [3] constructed a minimal asymptotic basis A with d(A) = 1/h. For
related research, one may refer to Chen and Chen [1], Chen and Tang [2], Jańczak and Schoen [5],
Nathanson [7, 8], Sun [11] and Tang and Lin [12].
Nathanson and Sárközy [9] proved the following results:
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
46 Ji-Zhen Xu and Yong-Gao Chen
Theorem A. If A is an asymptotic basis of order h and B is a subset of A with dL(B) > 1/h, then
there exists a finite subset F of A such that F ∪B is an asymptotic basis of order h.
Theorem B. If A is a minimal asymptotic basis of order h, then dL(A) ≤ 1/h.
In this paper, the following results are proved.
Theorem 1. Let h ≥ 2 be an integer. If A is an asymptotic basis of order h and B is a nonempty
subset of A, then either there exists a finite subset F of A such that F ∪B is an asymptotic basis of
order h, or for any ε> 0, there exists a finite subset Fε of A such that dL(h(Fε∪B)) ≥ hdL(B)−ε.
Remark. Theorem A is a corollary of Theorem 1. Let A be an asymptotic basis of order h and B1
a subset of A with dL(B1) > 1/h. We take ε= (hdL(B1)−1)/2. Then hdL(B1)−ε= (hdL(B1)+1)/2 >
1 ≥ dL(h(E ∪B1)) for any finite subset E of A. By Theorem 1, there exists a finite subset F of A
such that F ∪B1 is an asymptotic basis of order h.
Corollary 2. Let h ≥ 2 be an integer and let B be a nonempty set of nonnegative integers. Then
either there exists a finite set F of nonnegative integers such that F ∪B is an asymptotic basis of
order h, or for any ε> 0, there exists a finite set Fε of nonnegative integers such that dL(h(Fε∪B)) ≥
hdL(B)−ε.
Theorem 3. Let h ≥ 2 be an integer. If A is a minimal asymptotic basis of order h and B is a
nonempty subset of A, then for any ε> 0, there exists a finite subset Fε of A such that dL(h(Fε∪B)) ≥
hdL(B)−ε.
Theorem 4. Let h ≥ 2 be an integer. If A is a set of nonnegative integers with dL(A) > 0, then there
exists a subset B of A with dL(B) > 0 such that F ∪B is not an asymptotic basis of order h for any
finite set F .
2. Proofs
We will use a well known result of Kneser. If two sets X and Y of nonnegative integers are coincide
from some point on, then we write X ∼ Y . For any set X of nonnegative integers and any positive
integer g , let X (g ) be the set of all nonnegative integers n with n ≡ x (mod g ) for some x ∈ X .
In 1953, Kneser [6] proved the following profound result.
Lemma 5 (Kneser [6]). Let h ≥ 2 be an integer and X a nonempty set of nonnegative integers.
Then either dL(hX ) ≥ hdL(X ) or there exists a positive integer g such that hX ∼ hX (g ) and
dL(hX ) ≥ hdL(X )− h −1
g
.
Proof of Theorem 1. If there exists a finite subset F of A such that F ∪B is an asymptotic basis
of order h, then we are done. Now we assume that for any finite subset F of A, F ∪B is not an
asymptotic basis of order h. Let ε > 0. For any positive integer g , let Ag = {ag ,1, . . . , ag ,sg } be a
subset of A such that for every a ∈ A, there exists 1 ≤ i ≤ sg with a ≡ ag ,i (mod g ). It is clear that
A(g )g = A(g ). Let
Fε =
⋃
1≤g<(h−1)/ε
Ag .
Then Fε is finite. It is enough to prove that
dL(h(Fε∪B)) ≥ hdL(B)−ε.
By Lemma 5, either
dL(h(Fε∪B)) ≥ hdL(Fε∪B),
Ji-Zhen Xu and Yong-Gao Chen 47
or there exists a positive integer g1 such that h(Fε∪B) ∼ h((Fε∪B)(g1)) and
dL(h(Fε∪B)) ≥ hdL(Fε∪B)− h −1
g1
.
Since Fε is finite, it follows that dL(Fε∪B) = dL(B). Hence, either
dL(h(Fε∪B)) ≥ hdL(B),
or there exists a positive integer g1 such that h(Fε∪B) ∼ h((Fε∪B)(g1)) and
dL(h(Fε∪B)) ≥ hdL(B)− h −1
g1
.
If
dL(h(Fε∪B)) ≥ hdL(B),
then we are done. Now we assume that there exists a positive integer g1 such that h(Fε ∪B) ∼
h((Fε∪B)(g1)) and
dL(h(Fε∪B)) ≥ hdL(B)− h −1
g1
.
If (h − 1)/g1 ≤ ε, then we are done. Now we assume that (h − 1)/g1 > ε. We will derive a
contradiction. By (h −1)/g1 > ε, we have g1 < (h −1)/ε. Thus,
A ⊆ A(g1) = A(g1)g1 ⊆ F
(g1)
ε ⊆ (Fε∪B)(g1).
Hence
h A ⊆ h((Fε∪B)(g1)) ⊆N0. (1)
Since A is an asymptotic basis of order h, we have h A ∼N0. It follows from (1) that h((Fε∪B)(g1)) ∼
N0. Noting that h(Fε∪B) ∼ h((Fε∪B)(g1)), we have h(Fε∪B) ∼N0. This means that Fε∪B is an
asymptotic basis of order h, a contradiction.
This completes the proof of Theorem 1. □
Proof of Corollary 2. Since N0 is an asymptotic basis of order h and B ⊆N0, Corollary 2 follows
from Theorem 1 immediately. □
Proof of Theorem 3. Since A is a minimal asymptotic basis of order h, it follows that h A ∼N0. So
dL(h A) = 1. By Theorem B, dL(B) ≤ dL(A) ≤ 1/h. Thus, hdL(B) ≤ 1. Let ε> 0. If A \B is finite, then
for Fε = A \ B ,
dL(h(Fε∪B)) = dL(h A) = 1 ≥ hdL(B)−ε.
Now we assume that A \ B is infinite. Thus, for any finite subset F of A, we have F ∪B ̸= A. Since
A is a minimal asymptotic basis of order h, it follows that for any finite subset F of A, F ∪B is not
an asymptotic basis of order h. Now Theorem 3 follows from Theorem 1 immediately. □
Proof of Theorem 4. Let
B =
∞⋃
n=0
((
(h +1)n2+1, (h +1)(n+1)2
)
∩ A
)
.
For a sufficiently large x, let k be the integer with
(h +1)k2 ≤ x < (h +1)(k+1)2 .
Let t be the integer with
(h +1)t−1 < h(h +1)+dL(A)
(h +1)dL(A)
≤ (h +1)t .
48 Ji-Zhen Xu and Yong-Gao Chen
It is clear that t ≥ 1. If x ≤ (h +1)k2+t , then
B(x)
x
≥ B((h +1)
k2 )
x
≥ 1
(h +1)t
B((h +1)k2 )
(h +1)k2
> dL(A)
h(h +1)+dL(A)
B((h +1)k2 )
(h +1)k2 .
Since
B((h +1)k2 ) ≥ A((h +1)k2 )−
k−1∑
n=0
(
(h +1)n2+1 − (h +1)n2 +1
)
≥ A((h +1)k2 )−k(h +1)(k−1)2+1,
it follows that
B((h +1)k2 )
(h +1)k2 ≥
A((h +1)k2 )
(h +1)k2 +o(1) ≥ dL(A)+o(1).
Hence
B(x)
x
≥ dL(A)
h(h +1)+dL(A)
(dL(A)+o(1)).
If x > (h +1)k2+t , then
B(x) ≥ A(x)−
k∑
n=0
(
(h +1)n2+1 − (h +1)n2 +1
)
= A(x)−h
k∑
n=0
(h +1)n2 −k −1
≥ A(x)−h(h +1)k2 −hk(h +1)(k−1)2 −k −1
> A(x)− h
(h +1)t x −
hk
(h +1)2k−1+t x −k −1.
It follows that
B(x)
x
≥ A(x)
x
− h
(h +1)t −o(1)
≥ dL(A)− h(h +1)dL(A)
h(h +1)+dL(A)
−o(1)
= dL(A)
2
h(h +1)+dL(A)
−o(1).
Combining the above arguments, we have
dL(B) ≥ dL(A)
2
h(h +1)+dL(A)
> 0.
Let F be a finite set of nonnegative integers. Then there exists a positive integer m such that
F ⊆ [0, (h +1)m2+1].
For any integer n > m, by the definition of B ,
[(h +1)n2 , (h +1)n2+1]∩ (F ∪B) = [(h +1)n2 , (h +1)n2+1]∩B =;.
Since
(h +1)n2+1 > h(h +1)n2 ,
it follows that (h+1)n2+1 ∉ h(F ∪B). Therefore, F ∪B is not an asymptotic basis of order h for any
finite set F .
This completes the proof of Theorem 4. □
Ji-Zhen Xu and Yong-Gao Chen 49
References
[1] F.-J. Chen, Y.-G. Chen, “On minimal asymptotic bases”, Eur. J. Comb. 32 (2011), no. 8, p. 1329-1335.
[2] Y.-G. Chen, M. Tang, “On a problem of Nathanson”, Acta Arith. 185 (2018), no. 3, p. 275-280.
[3] P. Erdős, M. B. Nathanson, “Minimal asymptotic bases with prescribed densities”, Ill. J. Math. 32 (1988), no. 3, p. 562-
574.
[4] E. Härtter, “Ein Beitrag zur Theorie der Minimalbasen”, J. Reine Angew. Math. 196 (1956), p. 170-204.
[5] M. Jańczak, T. Schoen, “Dense minimal asymptotic bases of order 2”, J. Number Theory 130 (2010), no. 3, p. 580-585.
[6] M. Kneser, “Abschätzung der asymptotischen Dichte von Summenmengen”, Math. Z. 58 (1953), p. 459-484.
[7] M. B. Nathanson, “Minimal bases and maximal nonbases in additive number theory”, J. Number Theory 6 (1974),
p. 324-333.
[8] ——— , “Minimal bases and powers of 2”, Acta Arith. 51 (1988), no. 5, p. 95-102.
[9] M. B. Nathanson, A. Sárközy, “On the maximum density of minimal asymptotic bases”, Proc. Am. Math. Soc. 105
(1989), no. 1, p. 31-33.
[10] A. Stöhr, “Gelöste und ungelöste Fragen über Basen der natüurlichen Zahlenreihe”, J. Reine Angew. Math. 194 (1955),
p. 111-140.
[11] C.-F. Sun, “On a problem of Nathanson on minimal asymptotic bases”, J. Number Theory 218 (2021), p. 152-160.
[12] M. Tang, D.-R. Ling, “On asymptotic bases and minimal asymptotic bases”, Colloq. Math. 170 (2022), no. 1, p. 65-77.


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Comptes Rendus
Mathématique
Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos
On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities
Volume 362 (2024), p. 1253-1260
Online since: 5 November 2024
https://doi.org/10.5802/crmath.638
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1253-1260
https://doi.org/10.5802/crmath.638
Research article / Article de recherche
Harmonic analysis / Analyse harmonique
On the sharpness of some quantitative
Muckenhoupt–Wheeden inequalities
Optimalité des inégalités quantitatives de
Muckenhoupt–Wheeden
Andrei K. Lerner ∗,a, Kangwei Li b, Sheldy Ombrosi c,d and Israel P.
Rivera-Ríos e
a Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
b Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin,
China
c Departamento de Análisis Matemático y Matemática Aplicada Universidad
Complutense, Spain
d Departamento de Matemática e Instituto de Matemática. Universidad Nacional del
Sur - CONICET Argentina
e Departamento de Análisis Matemático, Estadística e Investigación Operativa y
Matemática Aplicada. Facultad de Ciencias. Universidad de Málaga (Málaga, Spain).
E-mails: lernera@math.biu.ac.il, kli@tju.edu.cn, sombrosi@ucm.es,
israelpriverarios@uma.es
Abstract. In the recent work [Cruz-Uribe et al. (2021)] it was obtained that
|{x ∈Rd : w(x)|G( f w−1)(x)| >α}|≲
[w]2A1
α
∫
Rd
| f |dx
both in the matrix and scalar settings, where G is either the Hardy–Littlewood maximal function or any
Calderón–Zygmund operator. In this note we show that the quadratic dependence on [w]A1 is sharp. This
is done by constructing a sequence of scalar-valued weights with blowing up characteristics so that the
corresponding bounds for the Hilbert transform and maximal function are exactly quadratic.
Résumé. Dans le récent travail [Cruz-Uribe et al. (2021)], il a été démontré
|{x ∈Rd : w(x)|G( f w−1)(x)| >α}|≲
[w]2A1
α
∫
Rd
| f |dx
à la fois dans les contextes matriciel et scalaire, où G est soit la fonction maximale de Hardy-Littlewood ou tout
opérateur de Calderón-Zygmund. Dans cette note, nous démontrons que la dépendance quadratique par
rapport à [w]A1 est optimale. Cela est réalisé en construisant une séquence de poids à valeurs scalaires avec
des caractéristiques d’éclatements, de sorte que les bornes correspondantes à la transformation de Hilbert et
la fonction maximale soient exactement quadratiques.
Keywords. Matrix weights, quantitative bounds, endpoint estimates.
Mots-clés. Poids matriciel, borne quantitative, estimations de type faible (1,1).
2020 Mathematics Subject Classification. 42B20, 42B25.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1254 Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos
Funding. The first author was supported by ISF grant no. 1035/21. The second author was supported by the
National Natural Science Foundation of China through project numbers 12222114 and 12001400. The fourth
author was supported by Spanish Ministerio de Ciencia e Innovación grant PID2022-136619NB-I00 and by
Junta de Andalucía grant FQM-354.
Manuscript received 12 March 2024, accepted 14 April 2024.
1. Introduction
Recall that a non-negative locally integrable function w satisfies the A1 condition if there exists a
constant C > 0 such that for every cube Q ⊂Rd ,
1
|Q|
∫
Q
w ≤C essinf
Q
w.
The smallest constant C for which this property holds is denoted by [w]A1 .
In the 70s, Muckenhoupt and Wheeden [10] established weighted weak type (1,1) bounds of
the form
|{x ∈R : w(x)|T ( f w−1)(x)| >α}|≲ Cw
α
∫
R
| f |dx, (1)
where T is either the Hilbert transform H or the Hardy–Littlewood maximal operator M . They
showed as well that w ∈ A1 is a sufficient condition for those inequalities to hold, even though it
is not necessary. Muckenhoupt and Wheeden observed that (1) could be regarded as a first step
to settle inequalities of the form
u1−r
({
ur |T f | >α})≲ 1
α
∫
R
| f |w dx, (2)
where u, w are non-negative functions and r ∈ [0,1]. Note that for u = w , for r = 0 this inequality
is the standard weak type inequality and in the case r = 1 it reduces to (1). Their idea was
to combine (2) with interpolation with change of measures in order to obtain two weighted
estimates.
Pushing forward that idea, Sawyer [13] showed that
uv
({
M( f v)
v
>α
})
≲Cu,v
1
α
∫
R
| f |uv dx, (3)
where u, v ∈ A1. This estimate combined with interpolation with change of measures allowed
him to reprove Muckenhoupt’s maximal theorem.
Since the aforementioned papers a number of works have been devoted to estimates related
to the ones above, that are known in the literature as mixed weak type estimates. Some worth
mentioning are [3] where (3) is extended to higher dimensions and further operators such as
Calderón–Zygmund operators via extrapolation, or [9] where it is shown that u ∈ A1 and v ∈ A∞
is sufficient for (3) to hold.
In terms of quantitative estimates for Cu,v in (3), and up to very recently for Cw in (1), as we
will mention soon, not very much is known. Some results are provided in the aforementioned
work [9] or in some other papers such as [12] or [1] for Cu,v in (3). The purpose of this note is
to provide some insight on Cw in (1). However, before presenting our main result we would like
to connect this problem with the matrix weighted setting. We devote the following lines to that
purpose.
In the last years quantitative matrix weighted estimates have attracted the attention of a
number of authors. Up until now only few sharp quantitative results in the matrix weight
setting are known. Among them the sharp Lp (W ) bounds for the maximal operator [7], the
sharp L2(W ) bound for the square function [6], and also the sharp Lp (W ) bounds in terms
Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos 1255
of the [W ]Aq constants with 1 ≤ q < p obtained in [8] for the maximal operator, Calderón–
Zygmund operators and commutators. Very recently Domelevo, Petermichl, Treil and Volberg [4]
showed the sharpness of the L2(W ) bound by [W ]3/2A2 for Calderón–Zygmund operators obtained
previously in [11].
Making sense of endpoint matrix weighted estimates is a tricky problem. Quite recently, Cruz-
Uribe et al. [2] managed to obtain the first quantitative endpoint estimates in that setting. In
order to state this result, we first give several definitions.
Assume that W is a matrix weight, that is, W is an n×n self-adjoint matrix function with locally
integrable entries such that W (x) is positive definite for a.e. x ∈Rd . Define the operator norm of
W by
∥W (x)∥ := sup
e∈Cn :|e|=1
|W (x)e|.
We say that W ∈ A1 if
[W ]A1 := sup
Q
esssup
y∈Q
1
|Q|
∫
Q
∥W (x)W (y)−1∥dx <∞.
It is easy to see that the matrix A1 constant [W ]A1 coincides with [w]A1 when n = 1.
Given a matrix weight W , a vector-valued function f⃗ : Rd → Cn and a Calderón–Zygmund
operator T , define
TW f⃗ (x) :=W (x)T (W −1 f⃗ )(x).
Next, define the maximal operator by
MW f⃗ (x) := sup
Q∋x
1
|Q|
∫
Q
|W (x)W −1(y) f⃗ (y)|dy.
The operators above have the obvious interpretation in the scalar setting.
Theorem A ([2]). We have∣∣{x ∈Rd : |TW f⃗ (x)| >α}∣∣≲ [W ]2A1
α
∫
Rd
| f⃗ |dx, (4)
and the same holds for MW .
At this point we are in the position to state the main result of this note.
Theorem 1. In the scalar-valued setting the quadratic dependence on [w]A1 in (4) is sharp for Tw
and for Mw .
As a direct consequence of Lemma 2 below, this result shows the sharpness of [W ]2A1 in
Theorem A in the matrix setting as well.
An interesting phenomenon here is the contrast between the strong L2(W ) and the weak
L1(W ) bounds for Calderón–Zygmund operators. As we mentioned above, the recent work [4]
establishes the sharpness of [W ]3/2A2 in the matrix setting. Comparing this with the linear A2 bound
in the scalar case [5], we see that the sharp weighted L2 bounds for Calderón–Zygmund operators
are different in the matrix and scalar settings. However, Theorem 1 shows that the sharp weighted
weak L1 bounds are the same in both settings.
2. Proof of Theorem 1
2.1. Connection between the scalar and the matrix weighted estimates
Lemma 2. Assume that GW stands either for TW or for MW . Then if∣∣{x ∈Rd : |GW f⃗ (x)| >α}∣∣≤ cϕ([W ]A1) 1α
∫
Rd
| f⃗ |dx,
1256 Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos
we have that for every w ∈ A1∣∣{x ∈Rd : |Jw f (x)| >α}∣∣≤ cϕ([w]A1) 1α
∫
Rd
| f |dx,
where Jw stands, respectively, for wT ( f w−1) or for w M( f w−1).
Proof. Let w ∈ A1. It is clear that W = w In is a matrix A1 weight. Furthermore,
[W ]A1 = [w]A1 .
Now given a scalar function f , we build f⃗ = ( f ,0, . . . ,0)t . Note that for these choices of f⃗ and W ,
clearly ∣∣GW f⃗ (x)∣∣= |Jw f (x)|.
This ends the proof. □
2.2. Construction of a family of weights providing the lower bound
We prove Theorem 1 via the following result.
Theorem 3. For any integer N > 20, there exists a scalar weight w ∈ A1(R) satisfying the following
properties:
(1)
∫ 1
0 w = 1;
(2) [w]A1 ≃ N ;
(3) |{x ∈ (1,∞) : w(x) > x}|≳ N 2.
Observe that Theorem 3 implies Theorem 1 immediately because if we take f =χ[0,1], then for
x > 1,
M f (x) = 1
x
, H f (x) =
∫
f (y)
x − y dy >
1
x
.
Hence, if T is either M or H , then
∥wT f ∥L1,∞ ≥
∣∣{x ∈ (1,∞) : w(x)|T f (x)| > 1}∣∣
≥
∣∣∣∣{x ∈ (1,∞) : w(x) · 1x > 1
}∣∣∣∣
≳ N 2 ≃ [w]2A1∥ f ∥L1(w).
The rest of this section will be devoted to proving Theorem 3.
Proof of Theorem 3. For k = 2,3, . . . , N we denote Jk = [2k ,2k+1). We will split Jk into small
intervals. Set Ik = [2k ,2k + k) and Lk = Jk \ Ik = [2k + k,2k+1). Let L−k and L+k be the left and
right halves of Lk , respectively. Next we define (L
−
k )
1 to be the right half of L−k and (L
+
k )
1 the left
half of L+k . Then
• when (L−k ) j = [a jk ,b jk) is defined, let (L−k ) j+1 = [a j+1k ,b j+1k ) satisfy that
b j+1k = a
j
k ,
∣∣(L−k ) j+1∣∣= 12 ∣∣(L−k ) j ∣∣;
• when (L+k ) j = [c jk ,d jk ) is defined, let (L+k ) j+1 = [c j+1k ,d j+1k ) satisfy that
c j+1k = d
j
k ,
∣∣(L+k ) j+1∣∣= 12 ∣∣(L+k ) j ∣∣.
Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos 1257
The process is stopped when we have (L−k )
k−1 and (L+k )
k−1 defined, and we simply define
(L−k )
k =
[
2k +k,2k +k + |L
−
k |
2k−1
)
, (L+k )
k =
[
2k+1 − |L
+
k |
2k−1
,2k+1
)
.
Now we have split Jk into disjoint intervals, namely,
Jk = Ik ∪
k⋃
j=1
(L−k )
j ∪
k⋃
j=1
(L+k )
j .
I4 L4
(L−4 )
1 (L+4 )
1(L−4 )
2 (L+4 )
2
(L−4 )
3 (L+4 )
3
(L−4 )
4 (L+4 )
4
Figure 1. Component intervals of J4.
Define
wk = 2k+1χIk +
k∑
j=1
2 jχ(L−k )
j ∪(L+k ) j
and our weight on [0,2N+2] is
w(x) =

χ[0,4)(x)+∑Nk=2 wk (x), x ∈ [0,2N+1),
2N , x = 2N+1,
w(2N+2 −x), x ∈ [2N+1,2N+2].
Figure 2. Graph of w4
Finally we extend w(x) from [0,2N+2] toRperiodically with period 2N+2. Such a weight trivially
satisfies
∫ 1
0 w = 1. Moreover, since w(x) > x on Ik , we have∣∣{x ∈ (1,∞) : w(x) > x}∣∣≥ N∑
k=2
|Ik | =
N∑
k=2
k ≃ N 2.
Hence it remains to check that [w]A1 ≃ N . Since w is periodic on R and symmetrical on
[0,2N+2], it suffices to prove that
sup
I⊂[0,2N+1]
w(I )
|I |essinf
x∈I
w(x)
≃ N
(we use the standard notation w(E) = ∫E w).
1258 Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos
Figure 3. Joint graph of w4, w5 and w6.
Observe that |L−k | = |L+k | = 12 (2k −k). Further,∣∣(L−k ) j ∣∣= ∣∣(L+k ) j ∣∣= 12 j+1 (2k −k), j = 1, . . . ,k −1,
and ∣∣(L−k )k ∣∣= ∣∣(L+k )k ∣∣= 12k (2k −k).
Hence,
wk (Jk ) = 2k+1|Ik |+
k∑
j=1
2 j
(∣∣(L−k ) j ∣∣+ ∣∣(L+k ) j ∣∣)
= 2k+1k + (k −1)(2k −k)+2(2k −k) ≃ k2k .
From this, when I = [0,2N+1] we have
w(I )
|I | = 2
−(N+1)
(
4+
N∑
k=2
wk (Jk )
)
≃ 2−(N+1)
(
4+
N∑
k=2
k2k
)
≃ N = N essinf
x∈I
w(x).
Therefore, we are left to prove that for any I ⊂ [0,2N+1],
w(I )
|I | ≲ N essinfx∈I w(x). (5)
At this point we will make the following elementary observation. Our weight w is a step
function, and for each two adjacent intervals from its definition, the “jump" of w is at most 2.
Since the “jumps" are multiplicative we have the following.
Claim A. If I intersects at most m intervals from the definition of w, then
w(I )
|I | ≤ maxx∈I w(x) ≤ 2
m min
x∈I
w(x) = 2m essinf
x∈I
w(x).
In what follows we will prove (5) according to the size of I .
Case 1. |I | ≤ 4. In this case, note that in each Jk (k ≥ 2), (L−k )k−1, (L−k )k and (L+k )k−1, (L+k )k are the
smallest intervals, and∣∣(L−k )k−1∣∣= ∣∣(L−k )k ∣∣= ∣∣(L+k )k−1∣∣= ∣∣(L+k )k ∣∣= 1− k2k ≥ 12 .
Hence I intersects at most 9 intervals from the definition of w , and we are in position to apply
Claim A with m = 9.
Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos 1259
Case 2. |I | > 4. In this case, we may assume |I | ∈ (2k0 ,2k0+1] with some k0 ≥ 2. We may further
assume k0 < N −10 as otherwise
w(I )
|I | ≲ 2
−N w([0,2N+1]) ≃ N essinf
x∈I
w(x).
Case 2a. I ⊂ [0,2k0+10]. Then similarly to above,
w(I )
|I | < 2
−k0 w([0,2k0+10]) ≃ k0 essinf
x∈I
w(x).
Case 2b. I ̸⊂ [0,2k0+10]. Then I ⊂ [2k0+9,2N+1]. Denote by ck the center of Lk .
Case 2b-a. I contains some ck with k ≥ k0 +9. Then the estimate is trivial since I ⊂ (L−k )1 ∪ (L+k )1
and we apply Claim A with m = 2.
Case 2b-b. I does not contain any ck . In this case we may assume I ⊂ (cℓ,cℓ+1) for some
k0 +8 ≤ ℓ≤ N .
Suppose that I = [a,b] and a ∈ (L+
ℓ
) j for some j . If j ≤ ℓ−k0 −4, then∣∣(L+ℓ ) j+1∣∣= |L+ℓ |2−( j+1) = 2ℓ−ℓ2 j+2 > 2k0+1,
so that I will intersect at most (L+
ℓ
) j and (L+
ℓ
) j+1 and we again apply Claim A with m = 2.
If j ≥ ℓ−k0 −3, note that then
I ⊂
ℓ⋃
j=ℓ−k0−3
(L+ℓ )
j ∪
ℓ+1⋃
i=ℓ−k0−2
(L−ℓ+1)
i ∪ Iℓ+1.
Here i ≥ ℓ−k0 −2 since∣∣(L−ℓ+1)ℓ−k0−2∣∣= 2ℓ+1 − (ℓ+1)2ℓ−k0−1 > 2
ℓ
2ℓ−k0−1
= 2k0+1 ≥ ℓ(I ).
Hence we have
w(I )
|I |essinf
x∈I
w(x)
≤
∑ℓ
j=ℓ−k0−3 w
(
(L+
ℓ
) j
)+∑ℓ+1i=ℓ−k0−2 w((L−ℓ+1)i )+w(Iℓ+1)
2k0 2ℓ−k0−3
≲
∑ℓ
j=ℓ−k0−3 2
j ·2ℓ− j +∑ℓ+1i=ℓ−k0−2 2i ·2ℓ+1−i + (ℓ+1)2ℓ+2
2ℓ
≲ ℓ.
It remains to consider the case a ∈ Iℓ+1 ∪ L−ℓ+1. However, in this case we just need to discuss
whether b ∈ (L−
ℓ+1)
j with some j ≤ ℓ−k0 −3 or not, which is completely similar. This completes
the proof. □
Acknowledgement
We would like to thank the anonymous referee for his/her careful reading that helped improving
the presentation of the paper.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
1260 Andrei K. Lerner, Kangwei Li, Sheldy Ombrosi and Israel P. Rivera-Ríos
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