Comptes Rendus Mathématique

Shi-Qiang Chen

Csaba Sándor

Quan-Hui Yang

Comptes Rendus
Mathématique
Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang
On a problem of Nathanson related to minimal asymptotic bases of order h
Volume 362 (2024), p. 71-76
Online since: 2 February 2024
https://doi.org/10.5802/crmath.530
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 71-76
https://doi.org/10.5802/crmath.530
Research article / Article de recherche
Number theory / Théorie des nombres
On a problem of Nathanson related to
minimal asymptotic bases of order h
Shi-Qiang Chen a, Csaba Sándor b,c,d and Quan-Hui Yang ∗,e
a School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China
b Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and
Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
c Department of Computer Science and Information Theory, Budapest University of Technology and
Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
d MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111
Budapest, Hungary
e School of Mathematics and Statistics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
E-mails: csq20180327@163.com, csandor@math.bme.hu, yangquanhui01@163.com
Abstract. For integer h ≥ 2 and A ⊆ N, we define h A to be the set of all integers which can be written as a
sum of h, not necessarily distinct, elements of A. The set A is called an asymptotic basis of order h if n ∈ h A
for all sufficiently large integers n. An asymptotic basis A of order h is minimal if no proper subset of A
is an asymptotic basis of order h. For W ⊆ N, denote by F∗(W ) the set of all finite, nonempty subsets of
W . Let A(W ) be the set of all numbers of the form
∑
f ∈F 2 f , where F ∈ F∗(W ). In this paper, we give some
characterizations of the partitionsN=W1∪·· ·∪Wh with the property that A = A(W1)∪·· ·∪A(Wh ) is a minimal
asymptotic basis of order h. This generalizes a result of Chen and Chen, recent result of Ling and Tang, and
also recent result of Sun.
Keywords. Asymptotic bases, minimal asymptotic bases, binary representation.
2020 Mathematics Subject Classification. 11B34.
Funding. S.Q.Chen is supported by the National Natural Science Foundation of China, Grant No. 12301003,
the Anhui Provincial Natural Science Foundation (Grant No. 2308085QA02) and the University Natural
Science Research Project of Anhui Province (Grant No. 2022AH050171). C.S. was supported by the NKFIH
Grants No. K129335. Q.H.Yang is supported by the National Natural Science Foundation of China, Grant
No. 12371005.
Manuscript received 15 February 2023, revised 12 June 2023 and 26 June 2023, accepted 26 June 2023.
1. Introduction
LetN be the set of all nonnegative integers. For an integer h ≥ 2 and A ⊆N, we define
h A = {n : n = a1 +·· ·+ah , ai ∈ A, i = 1,2, . . . ,h}.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
72 Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang
The set A is called an asymptotic basis of order h if n ∈ h A for all sufficiently large integers n. An
asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order
h. This means that for any a ∈ A, the set Ea = h A \ h(A \ {a}) is infinite.
Let W be a nonempty subset of N. Denote by F∗(W ) the set of all finite, nonempty subsets of
W . Let A(W ) be the set of all numbers of the form
∑
f ∈F 2 f , where F ∈F∗(W ).
In 1988, Nathanson [8] gave a construction of minimal asymptotic bases of order h.
Theorem A. Let h ≥ 2 and let Wi = {n ∈ N : n ≡ i (mod h)} for i = 0,1, . . . ,h − 1. Let A =
A(W0)∪ A(W1)∪·· ·∪ A(Wh−1). Then A is a minimal asymptotic basis of order h.
Let h ≥ 2 and N = W1 ∪ ·· · ∪Wh be a partition; that is, Wi ∩W j = ; if i ̸= j . Nathanson also
posed the following open problem:
Problem 1. Characterise the partitionsN=W1 ∪·· ·∪Wh with the property that A = A(W1)∪·· ·∪
A(Wh) is a minimal asymptotic basis of order h.
In 2011, Chen and Chen [1] solved Problem 1 for h = 2. They proved the next theorem.
Theorem B. Let N = W1 ∪W2 be a partition with 0 ∈ W1 such that W1 and W2 are infinite. Then
A = A(W1)∪ A(W2) is a minimal asymptotic basis of order 2 if and only if either W1 contains no
consecutive integers or W2 contains consecutive integers or both.
In 2020, Ling and Tang [7] focus on Problem 1 for h = 3; they proved the following result:
Theorem C. For any i ∈ {0,1,2,3,4,5}, if W0 = {n ∈ N : n ≡ i , i + 1 (mod 6)}, W1 = {n ∈ N : n ≡
i +2, i +4 (mod 6)} and W2 = {n ∈N : n ≡ i +3, i +5 (mod 6)}, then A = A(W0)∪ A(W1)∪ A(W2) is
a minimal asymptotic basis of order three.
In 2021, Sun [11] gave a generalization of Theorem A.
Theorem D. Let h and t be two positive integers with h ≥ 2. Let
W j =
∞⋃
i=0
[i ht + j t , i ht + j t + t −1]
for j = 0,1, . . . ,h − 1. Then A = A(W0)∪ A(W1)∪ ·· · ∪ A(Wh−1) is a minimal asymptotic basis of
order h.
In 2011, Chen and Chen [1] gave the following sufficient condition for Problem 1.
Theorem E. Let h ≥ 2 and r be the least integer with r > logh/log2. Let N = W1 ∪ ·· ·∪Wh be a
partition such that each set Wi is infinite and contains r consecutive integers for i = 1, . . . ,h. Then
A = A(W1)∪·· ·∪ A(Wh) is a minimal asymptotic basis of order h.
For other related results about minimal asymptotic bases, see [2–5, 10, 12].
In this paper, we continue to focus on Problem 1. First, we give a generalization of Theorem C
and Theorem D. For W ⊆N, set W (x) = |{n ∈W : n ≤ x}|.
Theorem 2. Let h ≥ 2 be an integer and N = W1 ∪ ·· · ∪Wh be a partition such that each set
W j (1 ≤ j ≤ h) satisfies |W j (ht −1)| = t for infinitely many integers t . Then A = A(W1)∪·· ·∪A(Wh)
is a minimal asymptotic basis of order h.
Our second theorem is a generalization of Theorem E.
Theorem 3. Let h be an integer, h ≥ 2, andN=W1∪·· ·∪Wh be a partition such that each set Wi is
infinite for i = 1, . . . ,h, 0 ∈W1 and every Wi contains
⌈ log(h+1)
log2
⌉
consecutive integers for i = 2, . . . ,h.
Then A = A(W1)∪·· ·∪ A(Wh) is a minimal asymptotic basis of order h.
Remark 4. It is easy to see that
⌈ log(h+1)
log2
⌉
is the least integer greater than logh/log2.
There is a more limited variant of Problem 1, see [6, 9, 12]
Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang 73
2. Lemmas
Lemma 5 (see [8, Lemma 1]). Let N = W1 ∪ ·· · ∪ Wh , where Wi ̸= ; for i = 1, . . . ,h. Then
A = A(W1)∪·· ·∪ A(Wh) is an asymptotic basis of order h.
Lemma 6. Let w1, . . . , ws be s distinct nonnegative integers. If
s∑
i=1
2wi ≡
t∑
j=1
2x j (mod 2ws+1),
where 0 ≤ x1, . . . , xt < ws +1 are integers not necessarily distinct, then there exist nonempty disjoint
sets J1, . . . , Js of {1,2, . . . , t } such that
2wi = ∑
j∈Ji
2x j
for i = 1, . . . , s.
Proof. By the proof of Lemma 2 from [8], there exist nonempty subsets J1, . . . , Js of {1,2, . . . , t }
such that
2wi = ∑
j∈Ji
2x j
for i = 1, . . . , s.
The result is trivial for s = 1. Now we assume that s ≥ 2. Since there exists a subset J1 of
{1,2, . . . , t } such that
2w1 = ∑
j∈J1
2x j ,
it follows that ∑
2≤i≤s
2wi ≡ ∑
j∈{1,...,t }\J1
2x j (mod 2ws+1),
which implies that there exists a subset J2 of {1,2, . . . , t } \ J1 such that
2w2 = ∑
j∈J2
2x j ,
and so J1 ∩ J2 =;. Continuing this process, it follows that there exist nonempty disjoint subsets
J1, . . . , Js of {1,2, . . . , t } such that
2wi = ∑
j∈Ji
2x j ,
for i = 1, . . . , s.
This completes the proof of Lemma 6. □
3. Proof of Theorem 2
By Lemma 5, it follows that A is an asymptotic basis of order h. For any a ∈ A, there exists
j ∈ {1,2, . . . ,h} such that a ∈ A(W j ). Without loss of generality, we may assume that a ∈ A(W1).
Then there exists a set K ⊆W1 such that a =∑i∈K 2i . Since there exist infinitely many t such that
|W j (ht −1)| = t
for j = 1,2, . . . ,h, it follows that there exists an integer tn such that
tnh ≤ minK < tn+1h.
Let
nT = a +
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v +
h∑
j=2
∑
v∈W j ∩[tn+1h,T ]
2v ,
74 Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang
where T is an integer such that 2T > a. Next we shall prove that nT ∈ Ea . Note that K (tn+1h−1) ̸=
; and
nT =
∑
j∈K (tn+1h−1)
2 j +
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v +2tn+1hm
for some m ≥ 0. Let nT = b1 +b2 +·· ·+bh be any representation of nT as a sum of h elements of
A and let
bi =
∑
i∈Si
2i
for i = 1,2. . . ,h. Let
ci =
∑
j∈Si (tn+1h−1)
2 j
for i = 1,2, . . . ,h. Then
ci ≡ bi (mod 2tn+1h)
and
|Si (tn+1h −1)| ≤ tn+1
for i = 1,2, . . . ,h, which implies that∑
k∈K (tn+1h−1)
2k +
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v ≡ ∑
1≤i≤h
ci ≡
∑
1≤i≤h
∑
j∈Si (tn+1h−1)
2 j (mod 2tn+1h).
Let ∑
k∈K (tn+1h−1)
2k +
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v ≡
s∑
j=1
2x j (mod 2tn+1h),
where s is an integer such that s ≤ tn+1h. By Lemma 6, there exist nonempty disjoint subsets
J0, J1, . . . , Jtn+1(h−1) of {1,2, . . . , s} such that
2minK +
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v = ∑
j∈J0
2x j + ∑
j∈J1
2x j + ∑
j∈J2
2x j +·· ·+ ∑
j∈Jtn+1(h−1)
2x j .
Therefore,
1+ tn+1(h −1) ≤ 1+|J1|+ · · ·+ |Jtn+1(h−1)| ≤ s ≤ tn+1h,
and so
tn+1(h −1) ≤ |J1|+ · · ·+ |Jtn+1(h−1)| ≤ tn+1h −1. (1)
Since |W j (tn+1h−1)| = tn+1 for any j ≥ 2, it follows from (1) and Lemma 6 that for any j ≥ 2, there
exist w ∈W j , J ⊆ {1,2, . . . , s} and |J | = 1 such that
2w = ∑
j∈J
2x j ,
which implies that x j = w ∈W j , and so
{b1, . . . ,bh} ̸⊆
h⋃
j=1, j ̸=m
A(W j )
for any m ≥ 2. Renumbering the indexes, we always assume that
bi ∈ A(Wi ), i ≥ 2.
Then
b1 = a +
(
h∑
j=2
∑
v∈W j ∩[0,tn+1h−1]
2v +
h∑
j=2
∑
v∈W j ∩[tn+1h,T ]
2v − ∑
2≤i≤h
bi
)
.
Since the binary representation of b1 is unique, it follows that b1 = a, that is nT ∈ Ea . Noting that
T is infinite, we have that A is minimal.
This completes the proof of Theorem 2.
Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang 75
4. Proof of Theorem 3
By Lemma 5, it follows that A is an asymptotic basis of order h. For a ∈ A, we assume that
a ∈ A(Wi ). Let Ea = h A \ h(A \ {a}). Now we prove that Ea is infinite.
Let
nT = a +
∑
w∈[0,T ]\Wi
2w ,
where T is an integer with T > a such that each [0,T ]∩W j contains
⌈ log(h+1)
log2
⌉
consecutive integers
for j = 2, . . . ,h. To prove that nT ∈ Ea , it suffices to prove that if nT = a1 +a2 +·· ·+ah with ai ∈ A
for 1 ≤ i ≤ h, then there exists at least one ak = a.
We distinguish two cases according to whether i = 1 or 2 ≤ i ≤ h.
Case 1. i = 1. Suppose that there exists an integer j ≥ 2 such that
{a1, a2, . . . , ah} ⊆
⋃
1≤l≤h,l ̸= j
A(W j ).
Let
{
b +1,b +2, . . . ,b +⌈ log(h+1)log2 ⌉}⊆ [0,T ]∩W j . Then by Lemma 6 there exists
{a′1, . . . , a
′
h} ⊆
⋃
1≤l≤h,l ̸= j
A(W j )∪ {0}
such that
2b+1 +·· ·+2b+
⌈
log(h+1)
log2
⌉
= a′1 +·· ·+a′h .
Since a′i ∉ A(W j ) for i = 1, . . . ,h, we have
a′i ≤ 20 +21 +·· ·+2b = 2b+1 −1
for i = 1, . . . ,h. It follows that
2b+1 +·· ·+2b+
⌈
log(h+1)
log2
⌉
≤ h(2b+1 −1) < h2b+1,
that is 2
⌈
log(h+1)
log2
⌉
−1 < h, a contradiction. Hence, for any integer j ≥ 2, we have
{a1, a2, . . . , ah}⊈
⋃
1≤l≤h,l ̸= j
A(Wl ).
Renumbering the indexes, we may assume that ai ∈ A(Wi ) for i = 2,3, . . . ,h. It follows that
a1 = a +
∑
2≤ j≤h
( ∑
w∈[0,T ]∩W j
2w −a j
)
.
Since a ∈ A(W1), W1, . . . ,Wh are disjoint, and the binary representation of a1 is unique, we have
a1 = a. Therefore nT ∈ Ea , and Ea is infinite.
Case 2. i ≥ 2. Since nT is odd, therefore we may assume that a1 ∈ A(W1). Let 2 ≤ j ≤ h, j ̸= i .
Similar to the Case 1, we get that
{a1, . . . , ah}⊈
⋃
1≤k≤h,k ̸= j
A(Wk ).
Renumbering the indexes, we may assume that a j ∈ A(W j ) for j = 2,3, . . . , i − 1, i + 1, . . . ,h. It
follows that
ai = a +
∑
1≤ j≤h, j ̸=i
( ∑
w∈[0,T ]∩W j
2w −a j
)
.
Since a ∈ A(Wi ), W1, . . . ,Wh are disjoint, and the binary representation of ai is unique, we have
ai = a. Therefore nT ∈ Ea , and Ea is infinite.
This completes the proof.
76 Shi-Qiang Chen, Csaba Sándor and Quan-Hui Yang
Acknowledgements
We would like to thank the anonymous reviewer for the valuable and detailed comments, which
helps us to improve the manuscript.
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 for the natural numbers”, J. Number Theory 12 (1980), p. 154-
159.
[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 two”, J. Number Theory 130 (2010), no. 3, p. 580-
585.
[6] X. Jia, M. B. Nathanson, “A simple construction of minimal asymptotic bases”, Acta Arith. 52, no. 2, p. 95-101.
[7] D.-R. Ling, M. Tang, “Some remarks on minimal asymptotic bases of order three”, Bull. Aust. Math. Soc. 102 (2020),
no. 1, p. 21-30.
[8] M. B. Nathanson, “Minimal bases and powers of 2”, Acta Arith. 49 (1988), no. 5, p. 525-532.
[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 natürlichen Zahlenreihe II”, J. Reine Angew. Math. 194 (1955),
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[11] C.-F. Sun, “On a problem of Nathanson on minimal asymptotic bases”, J. Number Theory 218 (2021), p. 152-160.
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