3 Fixed a typo in first line.

Let $A$ be a set of positive integers and $A+A = \{a_1 + a_2 | a_1,a_2 \in \mathbb{N} A \}$. If $A+A$ contains all positive integers, $A$ is called a basis (of order 2) of the set of positive integers. A basis $A$ is called a minimal basis, if no proper subset of $A$ is a basis.

E.g. the set of all numbers with only "0"s and "1"s in ternary representation is a (even minimal) basis. This is somehow the discrete analogon to the well known fact that the sumset of two Cantor ternary sets is the closed interval [0,2].

I'm looking for a 'smallest' minimal basis of the set of positive integers ('smallest' in the sense of the lexikographic order - also other orders would be interesting, e.g. the order induced by sum of inverse squares of the elements of the basis).

There are a lot of articles about minimal asymptotic basis of the set of positive integers ("asymptotic" meaning that only every \textit{sufficiently large} natural number is the sum of two elements of the basis). But I'm struggling to get or find my question above answered.

ADDENDUM: Sorry for not having been clear, and thank you for your comments. I hope the following remarks will clarify things a bit: a) Yes, I consider only order 2. b) To make my intention clearer, lets consider the finite set $M_n = \{0, 1, 2, 3, \ldots, n-1\}$ and ask for a basis $A$ such that $M_n \subset A + A$. E.g. $A = \{0,1,3,4,9,10,12,13\}$ is a basis of $M_{27}$ (this is just the ternary construction mentionned above). In the finite case I'm looking for the basis with the smallest number of elements and in case of a tie (same number of elements), I prefer the basis which comes later in lexicographic order (i.e. I would prefer $\{0,1,3,4,10\}$ to $\{0,1,3,4,9\}$. My intention with the question was to ask this problem not for finite $M_n$, but for $M_{\infty} = \mathbb{N}$ (sorry also for the confusion whether to include "0" or not). Since the cardinality of the "ternary basis" scales with $n^{2/3}$ and the cardinality ofthe proposed basis of odd numbers is $n/2$, I regard the "ternary basis" as a better one. Perhaps the problem is still not well defined and/or there is no such "best" basis. This would also be helpful for me to know.

2 clarification and example

ADDENDUM: Sorry for not having been clear, and thank you for your comments. I hope the following remarks will clarify things a bit: a) Yes, I consider only order 2. b) To make my intention clearer, lets consider the finite set $M_n = \{0, 1, 2, 3, \ldots, n-1\}$ and ask for a basis $A$ such that $M_n \subset A + A$. E.g. $A = \{0,1,3,4,9,10,12,13\}$ is a basis of $M_{27}$ (this is just the ternary construction mentionned above). In the finite case I'm looking for the basis with the smallest number of elements and in case of a tie (same number of elements), I prefer the basis which comes later in lexicographic order (i.e. I would prefer $\{0,1,3,4,10\}$ to $\{0,1,3,4,9\}$. My intention with the question was to ask this problem not for finite $M_n$, but for $M_{\infty} = \mathbb{N}$ (sorry also for the confusion whether to include "0" or not). Since the cardinality of the "ternary basis" scales with $n^{2/3}$ and the cardinality ofthe proposed basis of odd numbers is $n/2$, I regard the "ternary basis" as a better one. Perhaps the problem is still not well defined and/or there is no such "best" basis. This would also be helpful for me to know.

1

# Minimal basis of set of positive integers

Let $A$ be a set of positive integers and $A+A = \{a_1 + a_2 | a_1,a_2 \in \mathbb{N} \}$. If $A+A$ contains all positive integers, $A$ is called a basis (of order 2) of the set of positive integers. A basis $A$ is called a minimal basis, if no proper subset of $A$ is a basis.

E.g. the set of all numbers with only "0"s and "1"s in ternary representation is a (even minimal) basis. This is somehow the discrete analogon to the well known fact that the sumset of two Cantor ternary sets is the closed interval [0,2].

I'm looking for a 'smallest' minimal basis of the set of positive integers ('smallest' in the sense of the lexikographic order - also other orders would be interesting, e.g. the order induced by sum of inverse squares of the elements of the basis).

There are a lot of articles about minimal asymptotic basis of the set of positive integers ("asymptotic" meaning that only every \textit{sufficiently large} natural number is the sum of two elements of the basis). But I'm struggling to get or find my question above answered.