Problem 14

Erdős Problem #14: declared status 'open'. Formalized: yes. Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon>0$ and large $N$$$\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}?$$Is it possible that$$\lvert \{1,\ldots,N\}\backslash B\rvert =o(N^{1/2})?$$ Prize: no. OEIS: A143824. Tags: additive combinatorics, number theory, sidon sets.