Entropy lower bounds and sum-product phenomena

[2604.20233] Entropy lower bounds and sum-product phenomena Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2604.20233 (math) [Submitted on 22 Apr 2026 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Entropy lower bounds and sum-product phenomena Authors: Lampros Gavalakis , Marcel K. Goh , Ioannis Kontoyiannis View PDF HTML (experimental) Abstract: Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$. Comments: 25 pages, including references. Corrected the errors in Lemmas 3.6 and 4.3 pointed out by Rupert Li Subjects: Combinatorics (math.CO) ; Information Theory (cs.IT) MSC classes: 94A17, 11B13 Cite as: arXiv:2604.20233 [math.CO] (or arXiv:2604.20233v3 [math.CO] for this version) https://doi.org/10.48550/arXiv.2604.20233 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Marcel Goh [ view email ] [v1] Wed, 22 Apr 2026 06:32:37 UTC (28 KB) [v2] Wed, 29 Apr 2026 16:10:16 UTC (28 KB) [v3] Fri, 28 Aug 2026 12:02:33 UTC (29 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-04 Change to browse by: cs cs.IT math math.IT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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