Problem Statement
Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if\[\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.\]How does $f(N)$ grow?
Categories:
Additive Combinatorics Sidon Sets
Progress
Considered by Erdős and Freud [ErFr91], who proved\[\left(\frac{2}{\sqrt{3}}+o(1)\right)N^{1/2} \leq f(N) \leq \left(2+o(1)\right)N^{1/2}.\](Although both bounds were already given by Erdős in [Er81h].) Note that $2/\sqrt{3}=1.15\cdots$. The lower bound is taking a genuine Sidon set $B\subset [1,N/3]$ of size $\sim N^{1/2}/\sqrt{3}$ and taking the union with $\{N-b : b\in B\}$. The upper bound was improved by Pikhurko [Pi06] to\[f(N) \leq \left(\left(\frac{1}{4}+\frac{1}{(\pi+2)^2}\right)^{-1/2}+o(1)\right)N^{1/2}\](the constant here is $=1.863\cdots$).The analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\sim N^{1/2}$, as proved by Cilleruelo.
See also [30], [819], and [864].
Source: erdosproblems.com/840 | Last verified: January 16, 2026