Tensor growth rate and exact one-dimensional obstructions for constrained differences

Definitions and mathematical statement

Fix a prime $p$ and a subset $D\subseteq\mathbb F_p$ containing $0$. For an integer $n\ge1$, write $D^n$ for its Cartesian power, considered inside the vector space $\mathbb F_p^n$. Call $A\subseteq\mathbb F_p^n$ $D$-progression-free if there are no vectors $x\in\mathbb F_p^n$ and $y\in D^n\setminus\{0\}$ for which \[ x,\quad x+y,\quad x+2y\in A. \] All coordinates and scalar multiples are computed in $\mathbb F_p$. The restriction is on the difference vector: different coordinates of $y$ may be different elements of $D$, and some may vanish. Define the extremal size \[ a_D(n)=\max\{|A|:A\subseteq\mathbb F_p^n\text{ is }D\text{-progression-free}\}. \] The problem asks for a characterization of the pairs $(p,D)$ for which there exists a real constant $C=C(p,D)<p$ such that $a_D(n)\le C^n$ for every positive integer $n$. The same $C$ must work in all dimensions. Thus a density estimate $a_D(n)/p^n\to0$ alone does not answer the question: the requested saving is exponential in $n$. The original algebraic formulation is retained also in characteristic two, where $x+2y=x$; no extra requirement that the three displayed points be distinct is imposed. For odd $p$, their distinctness follows from $y\ne0$. The set $D=\{0\}$ is included as a boundary case in the classification.

Short English statement

Which allowed coordinate differences force every progression-free subset of a finite vector space to have an exponential loss in size?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000881 (AIM-COMBINATORICS-0006), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, High-dimensional phenomena in discrete analysis, item 2.3. Original workshop question underlying AIM-COMBINATORICS-0006. http://aimpl.org/highdimdiscrete/2/.