I came across the following problem about representation functions:
Produce a set $A$ such that $r(n) > 0$ for all $n \in [1,N]$, but with $|A| \leq \sqrt{4N+1}$.
Note that r(n) = \left|\{(a, a'): a, a' \in A, n = a+a' \} \right|
I think $A = \{0,1,2 \}$ would work with the interval being $[1,4]$. Then $3 \leq \sqrt{17}$.
A second part of the question shows that one can prove that $|A| \leq \sqrt{N}$ if it satisfies the above conditions. But $3 > \sqrt{4} = 2$. Does this mean that my set $A$ is wrong?