Define the
This content will become publicly available on December 1, 2024
For any subset
 NSFPAR ID:
 10502706
 Publisher / Repository:
 Cambridge University Press
 Date Published:
 Journal Name:
 The Bulletin of Symbolic Logic
 Volume:
 29
 Issue:
 4
 ISSN:
 10798986
 Page Range / eLocation ID:
 626 to 655
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract Collatz map on the positive integers${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ by setting$\mathbb {N}+1 = \{1,2,3,\dots \}$ equal to${\operatorname {Col}}(N)$ when$3N+1$ N is odd and when$N/2$ N is even, and let denote the minimal element of the Collatz orbit${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ . The infamous$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $ Collatz conjecture asserts that for all${\operatorname {Col}}_{\min }(N)=1$ . Previously, it was shown by Korec that for any$N \in \mathbb {N}+1$ , one has$\theta> \frac {\log 3}{\log 4} \approx 0.7924$ for almost all${\operatorname {Col}}_{\min }(N) \leq N^\theta $ (in the sense of natural density). In this paper, we show that for$N \in \mathbb {N}+1$ any function with$f \colon \mathbb {N}+1 \to \mathbb {R}$ , one has$\lim _{N \to \infty } f(N)=+\infty $ for almost all${\operatorname {Col}}_{\min }(N) \leq f(N)$ (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$N \in \mathbb {N}+1$ adic cyclic group$3$ at high frequencies. This estimation is achieved by studying how a certain twodimensional renewal process interacts with a union of triangles associated to a given frequency.$\mathbb {Z}/3^n\mathbb {Z}$ 
Abstract Let
K be an imaginary quadratic field and a rational prime inert in$p\geq 5$ K . For a curve$\mathbb {Q}$ E with complex multiplication by and good reduction at$\mathcal {O}_K$ p , K. Rubin introduced ap adicL function which interpolates special values of$\mathscr {L}_{E}$ L functions ofE twisted by anticyclotomic characters ofK . In this paper, we prove a formula which links certain values of outside its defining range of interpolation with rational points on$\mathscr {L}_{E}$ E . 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
extension${\mathbb {Z}}_p$ of the unramified quadratic extension of$\Psi _\infty $ . Along the way, we present a theory of local points over${\mathbb {Q}}_p$ of the Lubin–Tate formal group of height$\Psi _\infty $ for the uniformizing parameter$2$ .$p$ 
Abstract We prove that the rational cohomology group
vanishes unless$H^{11}(\overline {\mathcal {M}}_{g,n})$ and$g = 1$ . We show furthermore that$n \geq 11$ is pure Hodge–Tate for all even$H^k(\overline {\mathcal {M}}_{g,n})$ and deduce that$k \leq 12$ is surprisingly well approximated by a polynomial in$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$ q . In addition, we use and its image under Gysin pushforward 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.$H^{11}(\overline {\mathcal {M}}_{1,11})$ 
Abstract The wellstudied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient
, as a Baily–Borel compactification of a ball quotient${\mathcal {M}}^{\operatorname {GIT}}$ , and as a compactified${(\mathcal {B}_4/\Gamma )^*}$ K moduli space. From all three perspectives, there is a unique boundary point corresponding to nonstable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ . The spaces${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ and${\mathcal {M}}^{\operatorname {K}}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact${\overline {\mathcal {B}_4/\Gamma }}$ not the case. Indeed, we show the more refined statement that and${\mathcal {M}}^{\operatorname {K}}$ are equivalent in the Grothendieck ring, but not${\overline {\mathcal {B}_4/\Gamma }}$ K 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. 
Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting
G be a countable discrete abelian group and be commuting endomorphisms whose images have finite indices, we show that$\phi _1, \phi _2, \phi _3: G \to G$ If
has positive upper Banach density and$A \subset G$ , then$\phi _1 + \phi _2 + \phi _3 = 0$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$\phi _1(A) + \phi _2(A) + \phi _3(A)$ and a recent result of the first author.$\mathbb {Z}$ For any partition
, there exists an$G = \bigcup _{i=1}^r A_i$ such that$i \in \{1, \ldots , r\}$ contains a Bohr set. This generalizes a result of the second and third authors from$\phi _1(A_i) + \phi _2(A_i)  \phi _2(A_i)$ to countable abelian groups.$\mathbb {Z}$ If
have positive upper Banach density and$B, C \subset G$ is a partition,$G = \bigcup _{i=1}^r A_i$ contains a Bohr set for some$B + C + A_i$ . This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss.$i \in \{1, \ldots , r\}$ All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices
, the upper Banach density of$[G:\phi _j(G)]$ A (in (1)), or the number of sets in the given partition (in (2) and (3)).