Problem Statement
Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in A$). Estimate the maximal possible size of $\lvert A\rvert$ - in particular, is it true that\[\lvert A\rvert \leq (1+o(1))\frac{2}{\sqrt{3}}N^{1/2}?\]
Categories:
Number Theory Sidon Sets Additive Combinatorics
Progress
A problem of Erdős and Freud, who prove that\[\lvert A\rvert \geq (1+o(1))\frac{2}{\sqrt{3}}N^{1/2}.\]This is shown by taking a genuine Sidon set $B\subset [1,N/3]$ of size $\sim N^{1/2}/\sqrt{3}$ and taking the union with $\{N-b : b\in B\}$.For the analogous question with $n=a-b$ they prove that $\lvert A\rvert\sim N^{1/2}$.
This is a weaker form of [840].
Source: erdosproblems.com/864 | Last verified: January 19, 2026