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

<figure><img src="image_1.png" /></figure>

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CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
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Research article / *Article de recherche*
Number theory / *Théorie des nombres*

# On subsets of asymptotic bases

Ji-Zhen Xu<sup>a, b</sup> and Yong-Gao Chen<sup>*</sup>,<sup>a</sup>

**Abstract.** Let $h \ge 2$ 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 $F \cup B$ is an asymptotic basis of order $h$, or for any $\epsilon > 0$, there exists a finite subset $F_{\epsilon}$ of $A$ such that $d_{L}(h(F_{\epsilon} \cup B)) \ge h d_{L}(B) - \epsilon$, where $d_{L}(X)$ denotes the lower asymptotic density of $X$ and $hX$ denotes the set of all $x_1 + \cdots + x_h$ with $x_i \in X$ ($1 \le i \le h$). 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

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Let $h \ge 2$ be an integer. For $A \subseteq \mathbb{N}_0$, let

$$hA = \{a_1 + \cdots + a_h : a_1, \ldots, a_h \in A\}.$$

We define

$$d_L(A) = \liminf_{x \to +\infty} \frac{A(x)}{x},$$

where $A(x)$ is the number of positive integers in $A$ which do not exceed $x$. Usually, $d_L(A)$ is called the *lower asymptotic density* of $A$. If

$$\lim_{x \to +\infty} \frac{A(x)}{x}$$

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 $hA$ 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 $h \ge 2$, 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:

<footnote><sup>a</sup> School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People's Republic of China</footnote>

<footnote><sup>b</sup> 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)</footnote>

<footnote>* Corresponding author.</footnote>
**Theorem A.** If A is an asymptotic basis of order h and B is a subset of A with $d_L(B) > 1/h$, then there exists a finite subset F of A such that $F \cup B$ is an asymptotic basis of order h.

**Theorem B.** If A is a minimal asymptotic basis of order h, then $d_L(A) \le 1/h$.

In this paper, the following results are proved.

**Theorem 1.** Let $h \ge 2$ 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 $F \cup B$ is an asymptotic basis of order h, or for any $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of A such that $d_L(h(F_\varepsilon \cup B)) \ge hd_L(B) - \varepsilon$.

**Remark.** Theorem A is a corollary of Theorem 1. Let A be an asymptotic basis of order h and $B_1$ a subset of A with $d_L(B_1) > 1/h$. We take $\varepsilon = (hd_L(B_1) - 1)/2$. Then $hd_L(B_1) - \varepsilon = (hd_L(B_1) + 1)/2 > 1 \ge d_L(h(E \cup B_1))$ for any finite subset E of A. By Theorem 1, there exists a finite subset F of A such that $F \cup B_1$ is an asymptotic basis of order h.

**Corollary 2.** Let $h \ge 2$ 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 $F \cup B$ is an asymptotic basis of order h, or for any $\varepsilon > 0$, there exists a finite set $F_\varepsilon$ of nonnegative integers such that $d_L(h(F_\varepsilon \cup B)) \ge hd_L(B) - \varepsilon$.

**Theorem 3.** Let $h \ge 2$ be an integer. If A is a minimal asymptotic basis of order h and B is a nonempty subset of A, then for any $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of A such that $d_L(h(F_\varepsilon \cup B)) \ge hd_L(B) - \varepsilon$.

**Theorem 4.** Let $h \ge 2$ be an integer. If A is a set of nonnegative integers with $d_L(A) > 0$, then there exists a subset B of A with $d_L(B) > 0$ such that $F \cup B$ 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 $X \sim Y$. For any set X of nonnegative integers and any positive integer g, let $X^{(g)}$ be the set of all nonnegative integers n with $n \equiv x \pmod g$ for some $x \in X$.

In 1953, Kneser [6] proved the following profound result.

**Lemma 5 (Kneser [6]).** Let $h \ge 2$ be an integer and X a nonempty set of nonnegative integers. Then either $d_L(hX) \ge hd_L(X)$ or there exists a positive integer g such that $hX \sim hX^{(g)}$ and

$$d_L(hX) \ge hd_L(X) - \frac{h-1}{g}.$$

**Proof of Theorem 1.** If there exists a finite subset F of A such that $F \cup B$ is an asymptotic basis of order h, then we are done. Now we assume that for any finite subset F of A, $F \cup B$ is not an asymptotic basis of order h. Let $\varepsilon > 0$. For any positive integer g, let $A_g = \{a_{g,1}, \dots, a_{g,s_g}\}$ be a subset of A such that for every $a \in A$, there exists $1 \le i \le s_g$ with $a \equiv a_{g,i} \pmod g$. It is clear that $A_g^{(g)} = A^{(g)}$. Let

$$F_{\varepsilon} = \bigcup_{1 \le g < (h-1)/\varepsilon} A_g.$$

Then $F_\varepsilon$ is finite. It is enough to prove that

$$d_L(h(F_\varepsilon \cup B)) \ge hd_L(B) - \varepsilon.$$

By Lemma 5, either

$$d_L(h(F_\varepsilon \cup B)) \ge hd_L(F_\varepsilon \cup B),$$
or there exists a positive integer $g_1$ such that $h(F_\epsilon \cup B) \sim h((F_\epsilon \cup B)^{(g_1)})$ and

$$d_L(h(F_\epsilon \cup B)) \geq h d_L(F_\epsilon \cup B) - \frac{h-1}{g_1}.$$

Since $F_\epsilon$ is finite, it follows that $d_L(F_\epsilon \cup B) = d_L(B)$. Hence, either

$$d_L(h(F_\epsilon \cup B)) \geq h d_L(B),$$

or there exists a positive integer $g_1$ such that $h(F_\epsilon \cup B) \sim h((F_\epsilon \cup B)^{(g_1)})$ and

$$d_L(h(F_\epsilon \cup B)) \geq h d_L(B) - \frac{h-1}{g_1}.$$

If

$$d_L(h(F_\epsilon \cup B)) \geq h d_L(B),$$

then we are done. Now we assume that there exists a positive integer $g_1$ such that $h(F_\epsilon \cup B) \sim h((F_\epsilon \cup B)^{(g_1)})$ and

$$d_L(h(F_\epsilon \cup B)) \geq h d_L(B) - \frac{h-1}{g_1}.$$

If $(h-1)/g_1 \le \epsilon$, then we are done. Now we assume that $(h-1)/g_1 > \epsilon$. We will derive a contradiction. By $(h-1)/g_1 > \epsilon$, we have $g_1 < (h-1)/\epsilon$. Thus,

$$A \subseteq A^{(g_1)} = A_{g_1}^{(g_1)} \subseteq F_{\epsilon}^{(g_1)} \subseteq (F_{\epsilon} \cup B)^{(g_1)}.$$

Hence

$$hA \subseteq h((F_\epsilon \cup B)^{(g_1)}) \subseteq \mathbb{N}_0. \qquad (1)$$

Since A is an asymptotic basis of order h, we have $hA \sim \mathbb{N}_0$. It follows from (1) that $h((F_\epsilon \cup B)^{(g_1)}) \sim \mathbb{N}_0$. Noting that $h(F_\epsilon \cup B) \sim h((F_\epsilon \cup B)^{(g_1)})$, we have $h(F_\epsilon \cup B) \sim \mathbb{N}_0$. This means that $F_\epsilon \cup B$ is an asymptotic basis of order h, a contradiction.

This completes the proof of Theorem 1.

<figure><img src="image_2.png" /></figure>

**Proof of Corollary 2.** Since $\mathbb{N}_0$ is an asymptotic basis of order h and $B \subseteq \mathbb{N}_0$, Corollary 2 follows from Theorem 1 immediately. □

<figure><img src="image_3.png" /></figure>

**Proof of Theorem 3.** Since A is a minimal asymptotic basis of order h, it follows that $hA \sim \mathbb{N}_0$. So $d_L(hA) = 1$. By Theorem B, $d_L(B) \le d_L(A) \le 1/h$. Thus, $hd_L(B) \le 1$. Let $\epsilon > 0$. If $A \setminus B$ is finite, then for $F_\epsilon = A \setminus B$,

$$d_L(h(F_\epsilon \cup B)) = d_L(hA) = 1 \geq hd_L(B) - \epsilon.$$

Now we assume that $A \setminus B$ is infinite. Thus, for any finite subset $F$ of A, we have $F \cup B \ne A$. Since A is a minimal asymptotic basis of order h, it follows that for any finite subset $F$ of A, $F \cup B$ is not an asymptotic basis of order h. Now Theorem 3 follows from Theorem 1 immediately. □

<figure><img src="image_4.png" /></figure>

**Proof of Theorem 4.** Let

$$B = \bigcup_{n=0}^{\infty} \left( \left( (h+1)^{n+1}, (h+1)^{(n+1)^2} \right) \cap A \right).$$

For a sufficiently large x, let k be the integer with

$$(h+1)^k \le x < (h+1)^{(k+1)^2}.$$

Let t be the integer with

$$(h+1)^{t-1} < \frac{h(h+1) + d_L(A)}{(h+1)d_L(A)} \le (h+1)^t.$$
It is clear that $t \ge 1$. If $x \le (h+1)^{k^2+t}$, then

$$ \begin{aligned} \frac{B(x)}{x} &\ge \frac{B((h+1)^{k^2})}{x} \ge \frac{1}{(h+1)^t} \frac{B((h+1)^{k^2})}{(h+1)^{k^2}} \\ &> \frac{d_L(A)}{h(h+1) + d_L(A)} \frac{B((h+1)^{k^2})}{(h+1)^{k^2}}. \end{aligned} $$

Since

$$ \begin{aligned} B((h+1)^{k^2}) &\ge A((h+1)^{k^2}) - \sum_{n=0}^{k-1} ((h+1)^{n^2+1} - (h+1)^{n^2} + 1) \\ &\ge A((h+1)^{k^2}) - k(h+1)^{(k-1)^2+1}, \end{aligned} $$

it follows that

$$ \frac{B((h+1)^{k^2})}{(h+1)^{k^2}} \ge \frac{A((h+1)^{k^2})}{(h+1)^{k^2}} + o(1) \ge d_L(A) + o(1). $$

Hence

$$ \frac{B(x)}{x} \ge \frac{d_L(A)}{h(h+1) + d_L(A)} (d_L(A) + o(1)). $$

If $x > (h+1)^{k^2+t}$, then

$$ \begin{aligned} B(x) &\ge A(x) - \sum_{n=0}^{k} ((h+1)^{n^2+1} - (h+1)^{n^2} + 1) \\ &= A(x) - h \sum_{n=0}^{k} (h+1)^{n^2} - k - 1 \\ &\ge A(x) - h(h+1)^{k^2} - h k (h+1)^{(k-1)^2} - k - 1 \\ &> A(x) - \frac{h}{(h+1)^t} x - \frac{h k}{(h+1)^{2k-1+t}} x - k - 1. \end{aligned} $$

It follows that

$$ \begin{aligned} \frac{B(x)}{x} &\ge \frac{A(x)}{x} - \frac{h}{(h+1)^t} - o(1) \\ &\ge d_L(A) - \frac{h(h+1)d_L(A)}{h(h+1) + d_L(A)} - o(1) \\ &= \frac{d_L(A)^2}{h(h+1) + d_L(A)} - o(1). \end{aligned} $$

Combining the above arguments, we have

$$ d_L(B) \ge \frac{d_L(A)^2}{h(h+1) + d_L(A)} > 0. $$

Let $F$ be a finite set of nonnegative integers. Then there exists a positive integer $m$ such that

$$ F \subseteq [0, (h+1)^{m^2+1}]. $$

For any integer $n > m$, by the definition of $B$,

$$ [(h+1)^{n^2}, (h+1)^{n^2+1}] \cap (F \cup B) = [(h+1)^{n^2}, (h+1)^{n^2+1}] \cap B = \emptyset. $$

Since

$$ (h+1)^{n^2+1} > h(h+1)^{n^2}, $$

it follows that $(h+1)^{n^2+1} \notin h(F \cup B)$. Therefore, $F \cup B$ is not an asymptotic basis of order $h$ for any finite set $F$.

This completes the proof of Theorem 4.

□
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.