Sharp lower bounds for entropic subspace approximation

Definitions and mathematical statement

For a random variable $X$ taking values in a finite abelian group, put $H(X)=-\sum_x\mathbb P(X=x)\log_2\mathbb P(X=x)$, with $0\log_2 0=0$. For distributions $X,Y$ on the same group, their entropic Ruzsa distance is \[ d[X,Y]=H(X'-Y')-\tfrac12 H(X)-\tfrac12 H(Y), \] where $X',Y'$ are independent variables having the indicated distributions. In particular, the two copies used to define $d[X,X]$ are independent. For a linear subspace $V\leq\mathbb F_2^n$, let $U_V$ have the uniform distribution on $V$.

To express a lower bound that must survive the choice of the best subspace, define, for $K>1$, \[ F(K)=\sup_{n\geq1}\ \sup_{X:\,d[X,X]\leq\log_2 K} \min_{V\leq\mathbb F_2^n}d[X,U_V]. \] The workshop theorem supplies the upper estimate $F(K)\leq11\log_2 K$. The problem is to obtain sharp lower bounds on $F(K)$, and thereby determine how small the universal coefficient of $\log_2 K$ can be in such approximation theorems. A lower-bound example consists of a dimension and a distribution for which every subspace has the required separation; the subspace may depend on the distribution. Uniform distributions on subspaces can have distance zero, so a positive lower bound is necessarily an extremal assertion over $X$, rather than an assertion about every admissible variable. Translations of subspaces do not change this distance, so restricting the comparison distributions to linear subspaces does not lose affine translates.

Short English statement

How far can a distribution with small entropic doubling remain from every uniform subspace distribution?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000876 (AIM-COMBINATORICS-0001), 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 1.1. Original workshop question underlying AIM-COMBINATORICS-0001. http://aimpl.org/highdimdiscrete/1/.