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1. On Bertrand's and Rodriguez Villegas' higher-dimensional Lehmer conjecture, with an appendix by Gernando Rodriguez Villegas

   Ted Chinburg, Eduardo Friedman (January 2023, Pacific journal of mathematics)

   Let L be a number field and let E ⊂ OL∗ be any subgroup of the units of L. If rankZ(E) = 1, Lehmer’s conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rankZ(E) = rankZ(OL∗), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L : Q]. By sharpening a 1997 conjecture of Daniel Bertrand’s, Fernando Rodriguez Villegas “interpolated” between these two extremes of rank with a new higher-dimensional version of Lehmer’s conjecture. Here we prove a high-rank case of the Bertrand-Rodriguez Villegas conjecture. Namely, it holds if L contains a subfield K for which [L : K] ≫ [K : Q] and E contains the kernel of the norm map from OL∗ to OK∗ . 

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2. The factorisation property of l ^∞ ( X_k )

   https://doi.org/10.1017/s0305004120000304  

   LECHNER, RICHARD; MOTAKIS, PAVLOS; MÜLLER, PAUL F.X.; SCHLUMPRECHT, THOMAS (September 2021, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by $${(e_{(k,j)}^*)_j}$$ , for $$k\in\N$$ , let $$Z=\ell^\infty(X_k:k\kin\N)$$ be their l ∞ -sum, and let $$T:Z\to Z$$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 

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3. Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

   https://doi.org/10.1093/imrn/rny022  

   Bhattacharya, Bhaswar B; Ganguly, Shirshendu; Shao, Xuancheng; Zhao, Yufei (March 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract Let Xk denote the number of k-term arithmetic progressions in a random subset of $$\mathbb{Z}/N\mathbb{Z}$$ or $$\{1, \dots , N\}$$ where every element is included independently with probability p. We determine the asymptotics of $$\log \mathbb{P}\big (X_{k} \ge \big (1+\delta \big ) \mathbb{E} X_{k}\big )$$ (also known as the large deviation rate) where p → 0 with $$p \ge N^{-c_{k}}$$ for some constant ck > 0, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of p, the large deviation rate up to a constant factor. 

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4. On the rank of a random binary matrix

   https://doi.org/10.1137/1.9781611975482.58  

   Cooper, Colin; Frieze, Alan; Pegden, Wesley (January 2019, Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the rank of a random n × m matrix An, m; k with entries from GF(2), and exactly k unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all (nk) such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns m in terms of c, n, k, and where m = cn/k. The matrix An, m; k forms the vertex-edge incidence matrix of a k-uniform random hypergraph H. The rank of An, m; k can be expressed as follows. Let |C2| be the number of vertices of the 2-core of H, and | E (C2)| the number of edges. Let m* be the value of m for which |C2| = |E(C2)|. Then w.h.p. for m < m* the rank of An, m; k is asymptotic to m, and for m ≥ m* the rank is asymptotic to m – |E(C2)| + |C2|. In addition, assign i.i.d. U[0, 1] weights Xi, i ∊ 1, 2, … m to the columns, and define the weight of a set of columns S as X(S) = ∑j∊S Xj. Define a basis as a set of n – 1 (k even) linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for k = 2, the expected length of a minimum weight spanning tree tends to ζ(3) ∼ 1.202. 

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5. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

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