The entropic Marton-type conjecture for random variables over (F_2^m)

Let $c_1>0$ . For a random variable $X$ valued in $\mathbb{F}_2^m$ , let $X'$ be an independent copy of $X$ , and let $H$ denote Shannon entropy. For a subspace $G\leq\mathbb{F}_2^m$ , let $U_G$ be the uniform random variable on $G$ , independent of $X$ .

Entropic Marton-type conjecture. There exists $c_2=\exp(O(1/c_1))$ such that, for every $m\in\mathbb{Z}_+$ and every such $X$ , there is a subspace $G$ with

\dim G\leq c_2H(X)

and

H(U_G+X)-H(U_G)\leq (1+c_1)\bigl(H(X'+X)-H(X')\bigr).

The conjecture is intended to strengthen the paper's rate from $2^{-\Omega(\sqrt m)}$ to $2^{-\Omega(\sqrt m\log m)}$ , which would yield vanishing block-error probability up to Shannon capacity via the bit-to-block results cited by the authors. The statement is a conjectural entropic covering assertion related to the polynomial Freiman--Ruzsa and Marton-type phenomena; its resolution status is not specified in the supplied text.

A 2024 paper formulates the conjecture as an entropy-increment covering assertion over $\mathbb{F}_2^m$ , with a dimension bound controlled by $H(X)$ and a loss factor arbitrarily close to $1$ . The authors state that proving it would improve their coding-theoretic rate and yield vanishing block-error probability up to Shannon capacity.

Known results

Gowers, Green, Manners, and Tao proved the characteristic- $2$ polynomial Freiman–Ruzsa conjecture, yielding related entropic and covering statements with absolute constants.
A 2024 paper improved a related entropic Marton bound from $C'=11$ to $C'=10$ and obtained a covering exponent $C=9$ ; this is not the target assertion with arbitrary $c_1>0$ .

November 2024 explicit open statement

The directly relevant paper explicitly says that establishing this conjecture is left open. No retrieved source reports a proof, counterexample, verification, or subsequent status change.

Current status (as of September 2026): The exact entropic Marton-type conjecture remains open; related characteristic- $2$ Marton and polynomial Freiman–Ruzsa results are proved but do not settle its $(1+c_1)$ entropy bound with $c_2=\exp(O(1/c_1))$ .

The entropic Marton-type conjecture for random variables over $\mathbb{F}_2^m$

Primary source

Progress summary

Solutions 0