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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)

   Abstract 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. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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3. Fixed point sets and the fundamental group II: Euler characteristics

   https://doi.org/10.1017/prm.2023.75  

   Cappell, Sylvain; Weinberger, Shmuel; Yan, Min (October 2023, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   For a finite group$$G$$of not prime power order, Oliver showed that the obstruction for a finite CW-complex$$F$$to be the fixed point set of a contractible finite$$G$$-CW-complex is determined by the Euler characteristic$$\chi (F)$$. (He also has similar results for compact Lie group actions.) We show that the analogous problem for$$F$$to be the fixed point set of a finite$$G$$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on$$K_0$$[2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set. 

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4. IWASAWA THEORY FOR p -TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

   https://doi.org/10.1017/nmj.2022.30  

   BOOHER, JEREMY; CAIS, BRYDEN (June 2023, Nagoya Mathematical Journal)

   Abstract We investigate a novel geometric Iwasawa theory for$${\mathbf Z}_p$$-extensions of function fields over a perfect fieldkof characteristic$$p>0$$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$$\cdots \to X_2 \to X_1 \to X_0$$is the tower of curves overkassociated with a$${\mathbf Z}_p$$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$$X_n$$as$$n\rightarrow \infty $$. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$$X_n$$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$$k[V]$$-module structure of the space$$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$$of global regular differential forms as$$n\rightarrow \infty .$$For example, for each tower in a basic class of$${\mathbf Z}_p$$-towers, we conjecture that the dimension of the kernel of$$V^r$$on$$M_n$$is given by$$a_r p^{2n} + \lambda _r n + c_r(n)$$for allnsufficiently large, where$$a_r, \lambda _r$$are rational constants and$$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on$${\mathbf Z}_p$$-towers of curves, and we prove our conjectures in the case$$p=2$$and$$r=1$$. 

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5. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

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

   Abstract 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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