Stable Density on Subspaces

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.

Fix $0<\epsilon<1$. If $A\subset\mathbb F_2^n$ and $V\leq\mathbb F_2^n$ is a linear subspace, put $\alpha_A(V)=|A\cap V|/|V|$. Call $V$ $\epsilon$-stable for $A$ if \[ |\alpha_A(V)-\alpha_A(V')|\leq\epsilon \quad\text{for every linear }V'\leq V\text{ with }\dim V'=\dim V-1. \] The zero subspace is stable vacuously. Write $\operatorname{codim}(V)=n-\dim V$ and define the optimal uniform bound \[ D(\epsilon)=\sup_{n\geq1}\ \sup_{A\subset\mathbb F_2^n}\ \min\{\operatorname{codim}(V):V\text{ is }\epsilon\text{-stable for }A\}. \]

Statement recorded in the supplied source.

Obtain sharp bounds for $D(\epsilon)$ as $\epsilon\downarrow0$: how small a codimension, uniformly in $n$ and $A$, suffices to find a subspace whose density changes by at most $\epsilon$ on every codimension-one subspace? The source records an exponential-in-$1/\epsilon$ upper bound and power-of-$1/\epsilon$ lower examples; the question is to determine the correct quantitative dependence. [S2]

Short English statement

Determine how small a codimension guarantees a linear subspace on which the density of any given binary-vector-space set is stable under passing to every hyperplane.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 143 (GREEN-055). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 89 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.