Structure of Sets with Bounded Representation

Definitions and mathematical statement

For a positive integer $N$, write $[N]=\{1,\ldots,N\}$. Let $A\subset[N]$, let $c>0$ be a real parameter, and let $r\geq1$ be an integer parameter. Count a sum $x+y$ and the reversed sum $y+x$ as the same representation. With the usual convention that diagonal sums are also counted, this means \[ r_A(n)=|\{(x,y)\in A\times A:x\leq y,\ x+y=n\}|\qquad(n\in{\mathbb{Z}}). \] If the intended convention excludes $x=y$, one must replace $x\leq y$ by $x<y$; the supplied qualification explicitly resolves ordering but not the treatment of diagonal sums.

The hypotheses in the source are \[ |A|\geq c\sqrt N,\qquad \forall n\in{\mathbb{Z}},\quad r_A(n)\leq r. \] The requested research task is to determine what structural information can be deduced about $A$ from these hypotheses. The source does not specify a structural conclusion or equivalence relation, how $c$ and $r$ are quantified, whether they remain fixed as $N\to\infty$, or the permitted dependence of any conclusion on these parameters. Thus the displayed assumptions are precise under the stated representation convention, but they do not yet define a theorem to prove.

Under the diagonal-counting convention, the case $r=1$ is the Sidon condition that each sum has at most one unordered representation. The supplied literature triage dated 2026-08-17 reports that Eberhard--Manners give a projective-plane mechanism underlying all known dense Sidon constructions, classify the Desarguesian cases within that framework, exhibit many non-Desarguesian examples, and supply smaller algebraic examples. It reports their claim that all dense Sidon sets arise through this mechanism as a conjecture, not a classification theorem for arbitrary $A$ satisfying the displayed hypotheses. The source's exact meaning of "dense" and the construction mechanism are not defined in this record.

The registry marks the problem open, while its dated triage uses partially solved to describe construction-level progress. The same triage says Green presents the task as a vague structural programme and regards even formulating a conjecture as problematic. In particular, the background's assertion that all such sets have special structure is not a proved universal conclusion.

Short English statement

What structure is forced when a subset of $\{1,\ldots,N\}$ has at least $c\sqrt N$ elements but every integer has at most $r$ unordered representations as a sum of two of its elements? The source leaves the structural target and parameter quantifiers unspecified; known projective-plane constructions do not classify every such set.

Sources

• Sean Eberhard and Freddie Manners, "The apparent structure of dense Sidon sets," Electronic Journal of Combinatorics 30 (2023), no. 1, P1.33; arXiv:2107.05744: https://arxiv.org/abs/2107.05744.
• Ben Green, 100 Open Problems, Problem 34, most recent update December 2025 according to the supplied record: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The supplied triage attributes the programme's explicitly vague framing to this reference. The citation is retained without opening or generating a PDF.
• Local registry data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry, "Structure of Sets with Bounded Representation," in Ben Green's 100 Open Problems; statement, background, and literature triage dated 2026-08-17. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.