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Control and its applications in additive combinatorics

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arxiv 2501.09470 v1 pith:DRYPWMMM submitted 2025-01-16 math.NT math.CO

Thomas F. Bloom This is my paper · ORCID
classification math.NTmath.CO
keywords additiveapplicationsboundscombinatoricscontrolsetsarisesassumption
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abstract

We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growth of convex sets.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Entropic Sum-Product Phenomenon

    cs.IT 2026-07 accept novelty 8.0 of 10

    For every discrete real-valued random variable X with finite entropy, max{H(X+X'), H(XX')} ≥ (8/7)H(X) - O(log H(X)).

  2. Additive structure in convex sets

    math.CO 2025-09 conditional novelty 7.0 of 10

    Convex sets can contain Omega(|A|^{3/2}) three-term arithmetic progressions.

  3. Entropic additive energy and entropy inequalities for sums and products

    cs.IT 2025-06 conditional novelty 6.0 of 10

    Introduces differential entropic additive energy and proves new entropy bounds for sums, products, and sum-product combinations, plus a counterexample limiting the entropic Erdős-Szemerédi phenomenon to exponent at most 1/3.

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