On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups
<|reference_start|>On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups: We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another description: if a finite subset $A$ of an abelian group $G$ is such that the distribution of the sums $a+b$ with $(a,b) \in A \times A$ is only slightly more spread out than the uniform distribution on $A$, then $A$ has small symmetric difference with some finite coset of $G$. The resulting bounds are necessarily sharp up to a logarithmic factor.<|reference_end|>
@article{karam2024on,
    title={On entropy Marton-type inequalities and small symmetric differences with
  cosets of abelian groups},
    author={Thomas Karam},
    journal={arXiv preprint arXiv:2406.10872},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.10872},
    primaryClass={cs.IT math.CO math.GR math.IT math.NT math.PR}
}