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1. Almost all orbits of the Collatz map attain almost bounded values

   https://doi.org/10.1017/fmp.2022.8  

   Tao, Terence ( January 2022 , Forum of Mathematics, Pi)

   Define theCollatz map${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$on the positive integers$\mathbb {N}+1 = \{1,2,3,\dots \}$by setting${\operatorname {Col}}(N)$equal to$3N+1$whenNis odd and$N/2$whenNis even, and let${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$denote the minimal element of the Collatz orbit$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $. The infamousCollatz conjectureasserts that${\operatorname {Col}}_{\min }(N)=1$for all$N \in \mathbb {N}+1$. Previously, it was shown by Korec that for any$\theta> \frac {\log 3}{\log 4} \approx 0.7924$, one has${\operatorname {Col}}_{\min }(N) \leq N^\theta $for almost all$N \in \mathbb {N}+1$(in the sense of natural density). In this paper, we show that foranyfunction$f \colon \mathbb {N}+1 \to \mathbb {R}$with$\lim _{N \to \infty } f(N)=+\infty $, one has${\operatorname {Col}}_{\min }(N) \leq f(N)$for almost all$N \in \mathbb {N}+1$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$3$-adic cyclic group$\mathbb {Z}/3^n\mathbb {Z}$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.

    

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2. p -ADIC L -FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

   https://doi.org/10.1017/S147474802300021X  

   Burungale, Ashay A ; Kobayashi, Shinichi ; Ota, Kazuto ( May 2024 , Journal of the Institute of Mathematics of Jussieu)

   LetKbe an imaginary quadratic field and$p\geq 5$a rational prime inert inK. For a$\mathbb {Q}$-curveEwith complex multiplication by$\mathcal {O}_K$and good reduction atp, K. Rubin introduced ap-adicL-function$\mathscr {L}_{E}$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$\mathscr {L}_{E}$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE.

   A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic${\mathbb {Z}}_p$-extension$\Psi _\infty $of the unramified quadratic extension of${\mathbb {Q}}_p$. Along the way, we present a theory of local points over$\Psi _\infty $of the Lubin–Tate formal group of height$2$for the uniformizing parameter$-p$.

    

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3. The eleventh cohomology group of $\overline {\mathcal {M}}_{g,n}$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir ; Larson, Hannah ; Payne, Sam ( January 2023 , Forum of Mathematics, Sigma)

   We prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.

    

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4. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

   https://doi.org/10.1017/nmj.2023.27  

   Casalaina-Martin, Sebastian ; Grushevsky, Samuel ; Hulek, Klaus ; Laza, Radu ( June 2024 , Nagoya Mathematical Journal)

   The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient${(\mathcal {B}_4/\Gamma )^*}$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$are equivalent in the Grothendieck ring, but notK-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

    

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5. Bohr sets in sumsets II: countable abelian groups

   https://doi.org/10.1017/fms.2023.49  

   Griesmer, John T. ; Le, Anh N. ; Lê, Thái Hoàng ( July 2023 , Forum of Mathematics, Sigma)

   We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$\phi _1, \phi _2, \phi _3: G \to G$be commuting endomorphisms whose images have finite indices, we show that(1)

   If$A \subset G$has positive upper Banach density and$\phi _1 + \phi _2 + \phi _3 = 0$, then$\phi _1(A) + \phi _2(A) + \phi _3(A)$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$\mathbb {Z}$and a recent result of the first author.

   (2)

   For any partition$G = \bigcup _{i=1}^r A_i$, there exists an$i \in \{1, \ldots , r\}$such that$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$contains a Bohr set. This generalizes a result of the second and third authors from$\mathbb {Z}$to countable abelian groups.

   (3)

   If$B, C \subset G$have positive upper Banach density and$G = \bigcup _{i=1}^r A_i$is a partition,$B + C + A_i$contains a Bohr set for some$i \in \{1, \ldots , r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss.

   All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$[G:\phi _j(G)]$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)).

    

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