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Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $\boldsymbol{\Pi}^0_3$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in
Externí odkaz:
http://arxiv.org/abs/2305.10529
Autor:
Creiner, Alex, Jackson, Stephen
We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP with probab
Externí odkaz:
http://arxiv.org/abs/2210.12289
Given a target set $A\subseteq \mathbb{R}^d$ and a real number $\beta\in (0,1)$, McMullen introduced the notion of $A$ being an absolutely $\beta$-winning set. This involves a two player game which we call the $\beta$-McMullen game. We consider the v
Externí odkaz:
http://arxiv.org/abs/2110.03033
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the stri
Externí odkaz:
http://arxiv.org/abs/2108.03664
The Hausdorff $\delta$-dimension game was introduced by Das, Fishman, Simmons and {Urba{\'n}ski} and shown to characterize sets in $\mathbb{R}^d$ having Hausdorff dimension $\leq \delta$. We introduce a variation of this game which also characterizes
Externí odkaz:
http://arxiv.org/abs/2003.11578
We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$
Externí odkaz:
http://arxiv.org/abs/2003.02238
Akademický článek
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We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game and measu
Externí odkaz:
http://arxiv.org/abs/1910.11181
Autor:
Kaisinger, Lena R., Kentistou, Katherine A., Stankovic, Stasa, Gardner, Eugene J., Day, Felix R., Zhao, Yajie, Mörseburg, Alexander, Carnie, Christopher J., Zagnoli-Vieira, Guido, Puddu, Fabio, Jackson, Stephen P., O’Rahilly, Stephen, Farooqi, I. Sadaf, Dearden, Laura, Pantaleão, Lucas C., Ozanne, Susan E., Ong, Ken K., Perry, John R.B.
Publikováno v:
In Cell Genomics 9 August 2023 3(8)
Autor:
Chan, William, Jackson, Stephen
$\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. $\mathsf{ZF + AD} + \Theta$ is regular proves that for all nontriv
Externí odkaz:
http://arxiv.org/abs/1903.07005