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1. Sumsets and entropy revisited

   https://doi.org/10.1002/rsa.21252  

   Green, Ben; Manners, Freddie; Tao, Terence (July 2024, Random Structures & Algorithms)

   Abstract The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over . 

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2. An algorithmic hypergraph regularity lemma

   https://doi.org/10.1002/rsa.20739  

   Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias (December 2017, Random Structures & Algorithms)

   Abstract Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended tok‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs. 

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3. The Arinkin–Gaitsgory temperedness conjecture

   https://doi.org/10.1112/blms.12801  

   Færgeman, Joakim; Raskin, Sam (January 2023, Bulletin of the London Mathematical Society)

   Abstract Arinkin and Gaitsgory defined a category oftempered‐modules on that is conjecturally equivalent to the category of quasi‐coherent (not ind‐coherent!) sheaves on . However, their definition depends on the auxiliary data of a point of the curve; they conjectured that their definition is independent of this choice. Beraldo has outlined a proof of this conjecture that depends on some technology that is not currently available. Here we provide a short, unconditional proof of the Arinkin–Gaitsgory conjecture. 

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4. ASSOCIATIVITY OF THE MORLEY PRODUCT OF INVARIANT MEASURES IN NIP THEORIES

   https://doi.org/10.1017/jsl.2021.55  

   CONANT, GABRIEL; GANNON, KYLE (September 2021, The Journal of Symbolic Logic)

   Abstract In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories. 

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5. On a Conjecture by Hundertmark and Simon

   https://doi.org/10.1007/s00023-022-01169-x  

   Laptev, Ari; Loss, Michael; Schimmer, Lukas (May 2022, Annales Henri Poincaré)

   Abstract The main result of this paper is a complete proof of a new Lieb–Thirring-type inequality for Jacobi matrices originally conjectured by Hundertmark and Simon. In particular, it is proved that the estimate on the sum of eigenvalues does not depend on the off-diagonal terms as long as they are smaller than their asymptotic value. An interesting feature of the proof is that it employs a technique originally used by Hundertmark–Laptev–Weidl concerning sums of singular values for compact operators. This technique seems to be novel in the context of Jacobi matrices. 

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