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1. Theta functions, fourth moments of eigenforms and the sup-norm problem II

   https://doi.org/10.1017/fmp.2024.9  

   Khayutin, Ilya ; Nelson, Paul D ; Steiner, Raphael S ( January 2024 , Forum of Mathematics, Pi)

   Letfbe an$L^2$-normalized holomorphic newform of weightkon$\Gamma _0(N) \backslash \mathbb {H}$withNsquarefree or, more generally, on any hyperbolic surface$\Gamma \backslash \mathbb {H}$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$\mathbb {Q}$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

   with absolute implied constant. For a cuspidal Maaß newform$\varphi $of eigenvalue$\lambda $on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

   We establish analogous estimates in the setting of definite quaternion algebras.

    

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2. 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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3. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William ; Jackson, Stephen ; Trang, Nam ( January 2024 , Forum of Mathematics, Sigma)

   This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

   The following summarizes the main results proved under suitable partition hypotheses.•

   If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.

   •

   If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.

   •

   If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.

   •

   Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.

   •

   Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).

    

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4. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao ; Longbrake, Sean ; Ma, Jie ( February 2023 , Combinatorics, Probability and Computing)

   Given a family$\mathcal{F}$of bipartite graphs, theZarankiewicz number$z(m,n,\mathcal{F})$is the maximum number of edges in an$m$by$n$bipartite graph$G$that does not contain any member of$\mathcal{F}$as a subgraph (such$G$is called$\mathcal{F}$-free). For$1\leq \beta \lt \alpha \lt 2$, a family$\mathcal{F}$of bipartite graphs is$(\alpha,\beta )$-smoothif for some$\rho \gt 0$and every$m\leq n$,$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$(\alpha,\beta )$-smooth family$\mathcal{F}$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$is bipartite. In this paper, we strengthen their result by showing that for every real$\delta \gt 0$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\delta n^{\alpha -1}$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$\mathcal{F}$consisting of the single graph$K_{s,t}$when$t\gg s$. We also prove an analogous result for$C_{2\ell }$-free graphs for every$\ell \geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.

    

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5. 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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