Popular-direction planarization for no-five-coplanar sumsets

Definitions and mathematical statement

For a finite subset $A\subset\mathbb R^3$, define its sumset by $A+A=\{a+b:a,b\in A\}$, using ordinary vector addition and allowing $a=b$. Different ordered pairs yielding the same vector contribute only one sumset element. Say that $A$ satisfies the plane condition if every affine plane in $\mathbb R^3$ contains at most four distinct points of $A$. The condition concerns all affine planes, including planes not passing through the origin.

For each integer $n\geq5$, set \[ D_3(n)=\min\{|A+A|: A\subset\mathbb R^3, |A|=n,\ A\text{ satisfies the plane condition}\}. \] Such configurations exist, and the minimum is meaningful because the possible sumset sizes are integers in a finite range. The problem is to determine the asymptotic order of $D_3(n)$, or equivalently the largest universal growth factor $f(n)=D_3(n)/n$ in the inequality $|A+A|\geq n f(n)$ for these configurations. A sharp result should supply both a bound valid for every admissible set and constructions showing how small the doubling ratio can be. In particular, the requested factor depends only on cardinality; dependence on the individual set would not give a uniform extremal statement. No separation, bounded diameter, or integral-coordinate assumption is imposed. The workshop's comparison with planar sumsets is motivation rather than an additional hypothesis on $A$.

Short English statement

Determine how small the sumset of $n$ points in three-dimensional space can be when no five points lie in a plane.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000897 (AIM-COMBINATORICS-0022), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Additive combinatorics and its applications, item 2.05. Original workshop question underlying AIM-COMBINATORICS-0022. http://aimpl.org/addcombapp/2/.