Problem Statement
Let $A \subseteq \mathbb{N}$ and $B$ be integers representable in exactly one way as sum of two elements from $A$. Is $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2-\epsilon}$?
Categories:
Number Theory Sidon Sets Additive Combinatorics
Progress
Known Results
Erdos constructed a set $A$ where non-representable integers are $\ll_{\epsilon} N^{1/2+\epsilon}$, yet $\gg_{\epsilon} N^{1/3-\epsilon}$ for infinitely many $N$.
Erdos and Freud: For finite $A \subseteq \{1,\ldots,N\}$, non-uniquely representable integers $< 2^{3/2}N^{1/2}$.
Source: erdosproblems.com/14 | Last verified: January 13, 2026