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Let $S$ be a set of nonnegative integers such that the sumset $S+S$ is the nonnegative integers. If it can, what is the fastest growing function $f$ such that the number of elements of $S$ less than $n$ is proportional to $n/f(n)?$

$S$ can have zero natural density, with an example being the numbers that are sums of two nonnegative integer squares, but it has $f = \sqrt{\log {n}},$ so it looks suboptimal to me.

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    $\begingroup$ Maybe try something like the union of all the natural numbers with $0$ in the even entries and the natural numbers with $0$ in the odd entries (so, sticking to binary, $S$ contains numbers like $101010101$ and $100000$.). Clearly every natural number is a sum of two elements in $S$. $\endgroup$
    – lulu
    Commented Dec 22, 2022 at 18:08
  • $\begingroup$ The question is: what is its asymptotic density? $\endgroup$
    – mathlander
    Commented Dec 22, 2022 at 18:09
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    $\begingroup$ It is asymptotic to $\sqrt{n}.$ Solved. $\endgroup$
    – mathlander
    Commented Dec 22, 2022 at 18:12

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