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## 4. Asymptotic Bases

Let $A$ be a set of nonnegative integers. The representation function $r_A(n)$ counts the number of representations of an integer $n$ as the sum of $2$ elements of $A$: $r_A(n) = \text{card} \{ (a,a') \in A^2: a \leq a' \text{ and } a+a'=n\}.$ The sumset $2A = \{a+a': a,a' \in A\}$ is an asymptotic basis of order 2 if it contains all sufficiently large integers, or if, equivalently, the set $\mathbb{N}_0 \backslash 2A$ is finite.

An asymptotic basis $A$ is minimal if no proper subset of $A$ is an asymptotic basis of order 2. Equivalently, the set $A$ is a minimal asymptotic basis if, for every $a \in A$, the set $\{a' \in A: r_A(a+a') = 1\}$ is infinite.

##### Theorem.
(Nathanson [MR0347764]) Minimal asymptotic bases exist, but not every asymptotic basis contains a minimal asymptotic basis.

A simple example of an asymptotic basis of order 2 that does not contain a minimal asymptotic basis is $\{1\} \cup \{ 2n:n \in \mathbb{N}_0 \}$.

##### Theorem.
(Erdò‹s and Nathanson [MR0564925]) Let $A$ be a set of nonnegative integers. If $r_A(n) > c\log n$ for some $c > (\log(4/3))^{-1}$ and all $n \geq n_0$, then $A$ is an asymptotic basis of order 2 that contains a minimal asymptotic basis of order 2.

For background and many related problems see [MR0564925], [MR0894505], [MR0347764], [MR1023923].
1. #### Problem 4.1.

[Melvyn Nathanson] Let $A$ be an asymptotic basis of order 2 such that $r_A(n) > c\log n$ for some $c > 0$ and all $n \geq n_0$. Does $A$ contain a minimal asymptotic basis of order 2?
• #### Problem 4.2.

[Melvyn Nathanson] Let $A$ be an asymptotic basis of order 2 such that $\lim_{n\rightarrow \infty} r_A(n) = \infty.$ Does $A$ contain a minimal asymptotic basis of order 2?

Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.