arXiv subjects

Konstantinos A. Beros

Publications and source records attributed to Konstantinos A. Beros.

9 recordsLinked to original sources

arXiv2023

Maximal Tukey types, P-ideals and the weak Rudin-Keisler order

In this paper, we study some new examples of ideals on $ω$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order -- known as the "weak Rudin-Keisler order" -- and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin-Keisler. Acknowledgment: this preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in the Archive for Mathematical Logic, and is available online at https://doi.org/10.1007/s00153-023-00897-z.

math.LO↗

arXiv2020

Unilateral weighted shifts on $\ell^2$

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.

math.LO↗

arXiv2017

Canonical immunity and genericity

Whereas the usual notions of immunity -- e.g., immunity, hyperimmunity, etc. -- are associated with Cohen genericity, canonical immunity, as introduced by Beros, Khan and Kjos-Hanssen, is associated instead with Mathias genericity. Specifically, every Mathias generic is canonically immune and no Cohen 2-generic computes a canonically immune set.

math.LO↗

arXiv2016

Homomorphism reductions on Polish groups

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if $G$ is a Polish group and $H,L \subseteq G$ are subgroups, we say $H$ is {\em homomorphism reducible} to $L$ iff there is a continuous group homomorphism $φ: G \rightarrow G$ such that $H = φ^{-1} (L)$. We previously showed that there is a $K_σ$ subgroup, $L$, of the countable power of any locally compact Polish group, $G$, such that every $K_σ$ subgroup of $G^ω$ is homomorphism reducible to $L$. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.

math.LO↗

arXiv2016

Index Sets of Universal Codes

We examine sets of codes such that certain properties are invariant under the choice of oracle from a range of possible oracles and establish a connection between such codes and Medvedev reductions. In examing the complexity of such sets of \emph{universal codes}, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining $Π_1^1$-completeness for sets of universal codes. Among other corollaries, we show that the set of codes for Medvedev reductions of bi-immune sets to DNC functions is $Π_1^1$-complete.

math.LO↗

arXiv2015

Normal numbers and completeness results for difference sets

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes $\mathcal D_2 (\mathbfΠ^0_3)$ and $\mathcal D_ω(\mathbf Π^0_3)$, that is, the class of sets which are 2-differences (respectively, $ω$-differences) of $\mathbf Π^0_3$ sets.

math.LO↗

arXiv2014

Weak Rudin-Keisler reductions on projective ideals

We consider a slightly modified form of the standard Rudin-Keisler order on ideals and demonstrate the existence of complete (with respect to this order) ideals in various projective classes. Using our methods, we obtain a simple proof of Hjorth's theorem on the existence of a complete $\mathbf Π^1_1$ equivalence relation. Our proof of Hjorth's theorem enables us (under PD) to generalize his result to the classes of $\mathbf Π^1_{2n+1}$ equivalence relations.

math.LO↗

arXiv2014

Normal numbers and limit computable Cantor series

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Delta^0_{n+1} basic sequence with respect to which no Delta^0_n real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

math.LO↗

arXiv2013

Universal subgroups of Polish groups

Given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power of G has a universal K-sigma subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universal K-sigma and universal compactly generated subgroups. As an aside, we explore the relationship between the classes of K-sigma and compactly generated subgroups and give conditions under which the two coincide.

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