skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: On cohesive powers of linear orders
Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$$\omega $$,$$\zeta $$, and$$\eta $$denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$$\omega $$. If$$\mathcal {L}$$is a computable copy of$$\omega $$that is computably isomorphic to the usual presentation of$$\omega $$, then every cohesive power of$$\mathcal {L}$$has order-type$$\omega + \zeta \eta $$. However, there are computable copies of$$\omega $$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$$\omega + \zeta \eta $$. For example, we show that there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \eta $$. Our most general result is that if$$X \subseteq \mathbb {N} \setminus \{0\}$$is a Boolean combination of$$\Sigma _2$$sets, thought of as a set of finite order-types, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$, where$$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$denotes the shuffle of the order-types inXand the order-type$$\omega + \zeta \eta + \omega ^*$$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X)$$.  more » « less
Award ID(s):
2152095
PAR ID:
10518990
Author(s) / Creator(s):
; ; ; ; ;
Publisher / Repository:
Journal of Symbolic Logic
Date Published:
Journal Name:
The Journal of Symbolic Logic
Volume:
88
Issue:
3
ISSN:
0022-4812
Page Range / eLocation ID:
947 to 1004
Subject(s) / Keyword(s):
computable structures effective ultrapowers cohesive powers linear orders
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Forum of Mathematics, Sigma)
    Abstract 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$$\mathsf {DC}_{\mathbb {R}}$$). 
    more » « less
  2. (, Ergodic Theory and Dynamical Systems)
    Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if$$\Gamma $$is a countable discrete abelian group,$$\varphi , \psi \in \mathrm {End}(\Gamma )$$, and$$\psi - \varphi $$is an injective endomorphism with finite index image, then for any ergodic measure-preserving$$\Gamma $$-system$$( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$$, any measurable set$$A \in {\mathcal {X}}$$, and any$${\varepsilon }> 0$$, there is a syndetic set of$$g \in \Gamma$$such that$$\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$$. This generalizes the main results of Ackelsberget al[Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107]. For the group$$\Gamma = {\mathbb {Z}}^d$$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to thequasi-affine(orConze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to$$\varphi $$and$$\psi $$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. 
    more » « less
  3. ; ; (, Forum of Mathematics, Sigma)
    Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology. 
    more » « less
  4. ; (, Forum of Mathematics, Pi)
    Abstract Let$$\Omega _n$$be the ring of polynomial-valued holomorphic differential forms on complexn-space, referred to in physics as the superspace ring of rankn. The symmetric group$${\mathfrak {S}}_n$$acts diagonally on$$\Omega _n$$by permuting commuting and anticommuting generators simultaneously. We let$$SI_n \subseteq \Omega _n$$be the ideal generated by$${\mathfrak {S}}_n$$-invariants with vanishing constant term and study the quotient$$SR_n = \Omega _n / SI_n$$of superspace by this ideal. We calculate the doubly-graded Hilbert series of$$SR_n$$and prove an ‘operator theorem’, which characterizes the harmonic space$$SH_n \subseteq \Omega _n$$attached to$$SR_n$$in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki. 
    more » « less
  5. ; ; ; (, Forum of Mathematics, Pi)
    Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email