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Noting distinct
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JoshuaZ
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Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every sufficiently large integer is the sum of at most $k$ (not necessarily distinct) elements of $S$. Lagrange's four-square theorem can be thought of as a statement that the squares are an additive basis with $k=4$.

Given a set $S$, we will write $S^2= \{s^2: s \in S\}$.

Question: Is there an example of a set $S$ which is not an additive basis but where $S \cup S^2$ is an additive basis? (The same question then for asymptotic additive basis but I will not focus on that here.)

Note that any set with positive Schnirelmann density is an additive basis, so one naive way of solving this would be to exhibit a set $S$ which is not an additive basis but where $S \cup S^2$ has positive Schnirelmann density but this does not work; if $S$ has Schnirelmann density density zero then so will $S^2$.

Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every sufficiently large integer is the sum of at most $k$ elements of $S$. Lagrange's four-square theorem can be thought of as a statement that the squares are an additive basis with $k=4$.

Given a set $S$, we will write $S^2= \{s^2: s \in S\}$.

Question: Is there an example of a set $S$ which is not an additive basis but where $S \cup S^2$ is an additive basis? (The same question then for asymptotic additive basis but I will not focus on that here.)

Note that any set with positive Schnirelmann density is an additive basis, so one naive way of solving this would be to exhibit a set $S$ which is not an additive basis but where $S \cup S^2$ has positive Schnirelmann density but this does not work; if $S$ has Schnirelmann density density zero then so will $S^2$.

Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every sufficiently large integer is the sum of at most $k$ (not necessarily distinct) elements of $S$. Lagrange's four-square theorem can be thought of as a statement that the squares are an additive basis with $k=4$.

Given a set $S$, we will write $S^2= \{s^2: s \in S\}$.

Question: Is there an example of a set $S$ which is not an additive basis but where $S \cup S^2$ is an additive basis? (The same question then for asymptotic additive basis but I will not focus on that here.)

Note that any set with positive Schnirelmann density is an additive basis, so one naive way of solving this would be to exhibit a set $S$ which is not an additive basis but where $S \cup S^2$ has positive Schnirelmann density but this does not work; if $S$ has Schnirelmann density density zero then so will $S^2$.

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JoshuaZ
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Additive basis of a set union the square of the set

Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every sufficiently large integer is the sum of at most $k$ elements of $S$. Lagrange's four-square theorem can be thought of as a statement that the squares are an additive basis with $k=4$.

Given a set $S$, we will write $S^2= \{s^2: s \in S\}$.

Question: Is there an example of a set $S$ which is not an additive basis but where $S \cup S^2$ is an additive basis? (The same question then for asymptotic additive basis but I will not focus on that here.)

Note that any set with positive Schnirelmann density is an additive basis, so one naive way of solving this would be to exhibit a set $S$ which is not an additive basis but where $S \cup S^2$ has positive Schnirelmann density but this does not work; if $S$ has Schnirelmann density density zero then so will $S^2$.