Sidon Set Size Bounds

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A finite set $A\subset\mathbb Z$ is Sidon if $a+b=c+d$ with $a,b,c,d\in A$ implies equality of the unordered pairs $\{a,b\}=\{c,d\}$, with multiplicity retained when a pair repeats an element. Equivalently, each nonzero ordered difference has at most one representation. For $[N]=\{1,\ldots,N\}$, put \[ F(N)=\max\{|A|:A\subset[N]\text{ is Sidon}\}. \] The notation $O(1)$ denotes an absolute bounded term independent of $N$.

Statement recorded in the supplied source.

Improve lower or upper estimates for $F(N)$, at least along infinitely many integers $N\to\infty$, in comparison with the square-root benchmarks \[ \sqrt N+O(1)\leq F(N)\leq\sqrt N+N^{1/4}+O(1). \] The lower benchmark is realized along the prime-power construction sequence $N=q^2+q+1$ with $F(N)\geq q+1$; it is not being asserted uniformly for every integer $N$. The request includes improvements of the second-order term, not merely a proof of $F(N)=N^{1/2+o(1)}$. [S2]

Short English statement

Sharpen the known size estimates for Sidon subsets of an interval, even if the improvement holds only for an infinite sequence of interval lengths.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 162 (GREEN-074). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Daniel Carter, Zach Hunter and Kevin O’Bryant, On the Diameter of Finite Sidon Sets, arXiv:2310.20032v1 (2023), §1. It states the equivalent interval extremal problem and Singer and Erdős--Turán benchmarks, and gives the improved upper bound $F(N)\leq\sqrt N+0.98183N^{1/4}+O(1)$. https://arxiv.org/abs/2310.20032.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 31 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.