Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.
Marton's conjecture in abelian groups with bounded torsion
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abstract
We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let $G$ be an abelian group of torsion $m$ (meaning $mg=0$ for all $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \leq K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors.
years
2026 2representative citing papers
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Entropy lower bounds and sum-product phenomena
Entropy lower bounds are established for sums and products, including a max(H(X+X'), H(XX')) bounded below by a linear function of H(X) and min-entropy of X over arbitrary fields.
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AI technologies including ITPs and LLMs are transforming mathematical practice by altering proof verification, enabling trust-free large-scale collaboration, and redistributing epistemic labor between humans and machines.