Title: COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION
Abstract Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $ . Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $ . Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $ , $R(\ell ,D)$ implies $R(\ell ,C)$ , there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$ , $R(\ell ,\Lambda (\ell ))$ . In particular, under these assumptions, for all $n \in \omega $ , $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization. more »« less
Chan, William; Jackson, Stephen; Trang, Nam(
, 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}}$).
Wang, Ruosong; Woodruff, David P.(
, ACM Transactions on Algorithms)
An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications.
Abstract A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .
LEMPP, STEFFEN; MILLER, JOSEPH S.; NIES, ANDRÉ; SOSKOVA, MARIYA I.(
, The Journal of Symbolic Logic)
Abstract The tower number ${\mathfrak t}$ and the ultrafilter number $\mathfrak {u}$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of $\omega $ and the almost inclusion relation $\subseteq ^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory. We say that a sequence $(G_n)_{n \in {\mathbb N}}$ of computable sets is a tower if $G_0 = {\mathbb N}$ , $G_{n+1} \subseteq ^* G_n$ , and $G_n\smallsetminus G_{n+1}$ is infinite for each n . A tower is maximal if there is no infinite computable set contained in all $G_n$ . A tower ${\left \langle {G_n}\right \rangle }_{n\in \omega }$ is an ultrafilter base if for each computable R , there is n such that $G_n \subseteq ^* R$ or $G_n \subseteq ^* \overline R$ ; this property implies maximality of the tower. A sequence $(G_n)_{n \in {\mathbb N}}$ of sets can be encoded as the “columns” of a set $G\subseteq \mathbb N$ . Our analogs of ${\mathfrak t}$ and ${\mathfrak u}$ are the mass problems of sets encoding maximal towers, and of sets encoding towers that are ultrafilter bases, respectively. The relative position of a cardinal characteristic broadly corresponds to the relative computational complexity of the mass problem. We use Medvedev reducibility to formalize relative computational complexity, and thus to compare such mass problems to known ones. We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin’s characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e. set computes a maximal tower but no 1-generic $\Delta ^0_2$ -set does so. We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively.
Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M , g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.
CHAN, WILLIAM, JACKSON, STEPHEN, and TRANG, NAM. COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION. Retrieved from https://par.nsf.gov/biblio/10410126. The Journal of Symbolic Logic . Web. doi:10.1017/jsl.2022.78.
@article{osti_10410126,
place = {Country unknown/Code not available},
title = {COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION},
url = {https://par.nsf.gov/biblio/10410126},
DOI = {10.1017/jsl.2022.78},
abstractNote = {Abstract Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $ . Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $ . Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $ , $R(\ell ,D)$ implies $R(\ell ,C)$ , there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$ , $R(\ell ,\Lambda (\ell ))$ . In particular, under these assumptions, for all $n \in \omega $ , $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.},
journal = {The Journal of Symbolic Logic},
author = {CHAN, WILLIAM and JACKSON, STEPHEN and TRANG, NAM},
}
Warning: Leaving National Science Foundation Website
You are now leaving the National Science Foundation website to go to a non-government website.
Website:
NSF takes no responsibility for and exercises no control over the views expressed or the accuracy of
the information contained on this site. Also be aware that NSF's privacy policy does not apply to this site.
