arXiv subjects

Noam Greenberg

Publications and source records attributed to Noam Greenberg.

At least 19 recordsLinked to original sources

arXiv2026

Hyperarithmetic directions can all be exceptional for Marstrand's projection theorem

We show that there is a $\Pi^0_2$ subset $B$ of the Euclidean plane, of Hausdorff dimension 1, such that for every line $\ell$ through the origin with hyperarithmetic direction, the projection of $B$ onto $\ell$ has Hausdorff dimension 0, thus is exceptional for Marstrand's projection theorem. It follows that for any computable ordinal $\alpha$, being $\alpha$-random does not guarantee the ``almost-all'' statement of the projection theorem.

math.LO↗

arXiv2026

Characterising SJT reducibility

SJT reducibility between sets $A,B \subseteq \mathbb N$ is defined by $A \le_{SJT} B$ if for each computable function $h$ that is unbounded and nondecreasing, there is an $h$-bounded uniformly $B$-c.e.\ trace $(T_n)_{n \in \mathbb N} $ such that for each $n$, the value $J^A(n)$ of the jump is in $T_n$, if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the $K$-trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.

math.LO↗

arXiv2026

Forcing and classes of $\mathsf{HYP}$-dominating functions

We study the question, what computational power is sufficient to perform constructions using either Laver or Hechler forcing. As a result, we obtain a separation between three relativised non-lowness classes that are the computability-theoretic analogues of three of the cardinals in Cichon's diagram.

math.LO↗

arXiv2026

Topology, forcing, and graph colourings

We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.

math.GN↗

arXiv2025

Low$_2$ computably enumerable sets have hyperhypersimple supersets

A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal{L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal{L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e.\ $A$ is low$_2$ then $A$ has an atomless hyperhypersimple superset. In fact, if $A$ is c.e.\ and low$_2$, then for any $Σ_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$.

math.LO↗

arXiv2022

An Effective Classification of Borel Wadge Classes

We give a new and effective classification of all Borel Wadge classes of subsets of Baire space. This relies on the true stage machinery originally developed by Montalbán. We use this machinery to give a new proof of Louveau and Saint-Raymond's separation theorem for Borel Wadge classes. This gives a proof of Borel Wadge determinacy in the subsystem $\text{ATR}_0+Π^1_1$-I of second-order arithmetic.

math.LO↗

arXiv2022

Iterated Priority Arguments in Descriptive Set Theory

We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff-Kuratowski and Wadge theorems on the structure of ${\mathbf Δ}^0_ξ$, Louveau and Saint-Raymond's separation theorem, and Louveau's separation theorem.

math.LO↗

arXiv2022

Many forcing axioms for all regular uncountable cardinals

A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.

math.LO↗

arXiv2022

Martin-Löf reducibility and cost functions

Martin-Löf (ML)-reducibility compares $K$-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every $K$-trivial set is computable from a c.e.\ set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e.\ set ML-above the sets obeying it (called an ML-complete set for the cost function). We characterise the $K$-trivial sets computable from a fragment of the left-c.e.\ random real~$Ω$. This leads to a new characterisation of strong jump-traceability.

math.LO↗

arXiv2021

Cousin's lemma in second-order arithmetic

Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $\mathsf{ATR}_0$ (modulo some induction).

math.LO↗

arXiv2020

Computing sets from all infinite subsets

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $Π^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.

math.LO↗

arXiv2020

Realizing Computably Enumerable Degrees in Separating Classes

We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.

math.LO↗

arXiv2019

Highness properties close to PA-completeness

Suppose we are given a computably enumerable object arise from algorithmic randomness or computable analysis. We are interested in the strength of oracles which can compute an object that approximates this c.e. object. It turns out that, depending on the type of object, the resulting highness property is either close to, or equivalent to being PA-complete. We examine, for example, dominating a c.e. martingale by an oracle-computable martingale, computing compressions functions for two variants of Kolmogorov complexity, and computing subtrees of positive measure of a given $Π^0_1$ tree of positive measure without dead ends. We prove a separation result from PA-completeness for the latter property, called the \emph{continuous covering property}. We also separate the corresponding principles in reverse mathematics.

math.LO↗

arXiv2019

Bad oracles in higher computability and randomness

Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which $Ψ(A)\ne X$ for any higher functional~$Ψ$ which is consistent on all oracles. We construct an oracle~$A$ relative to which there is no universal higher ML-test. On the other hand, we show that badness has its limits: there are no higher self-PA oracles, and for no~$A$ can we construct a higher $A$-c.e.\ set which is also higher $A$-ML-random. We study various classes of bad oracles and differentiate between them using other familiar classes. For example, bad oracles for consistent reductions can be higher ML-random, whereas bad oracles for universal tests cannot.

math.LO↗

arXiv2019

Computing from projections of random points: a dense hierarchy of subideals of the $K$-trivial degrees

We study the sets that are computable from both halves of some (Martin-Löf) random sequence, which we call \emph{$1/2$-bases}. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e.\ elements. It is a proper subideal of the $K$-trivial sets. We characterise $1/2$-bases as the sets computable from both halves of Chaitin's $Ω$, and as the sets that obey the cost function $\mathbf c(x,s) = \sqrt{Ω_s - Ω_x}$. Generalising these results yields a dense hierarchy of subideals in the $K$-trivial degrees: For $k< n$, let $B_{k/n}$ be the collection of sets that are below any $k$ out of $n$ columns of some random sequence. As before, this is an ideal generated by its c.e.\ elements and the random sequence in the definition can always be taken to be $Ω$. Furthermore, the corresponding cost function characterisation reveals that $B_{k/n}$ is independent of the particular representation of the rational $k/n$, and that $B_p$ is properly contained in $B_q$ for rational numbers $p< q$. These results are proved using a generalisation of the Loomis--Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyse arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the $K$-trivial sets, we can calculate from the family which $B_p$ it characterises. We finish by showing that the the union of $B_p$ for $p<1$ is the collection of sets which are robustly computable from a random, a class previously studied by Hirschfeldt, Jockusch, Kuyper, and Schupp.

math.LO↗

arXiv2019

Relationships between computability-theoretic properties of problems

A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of $ω$ cones, of 1 hyperimmunity or of 1 non-$Σ^0_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non-$Σ^0_1$ definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.

math.LO↗

arXiv2017

Dimension 1 sequences are close to randoms

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly $s$-random sequence. Further, for any $s<t$, every sequence of effective dimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of its bits to produce a sequence of effective dimension $t$, and this bound is optimal.

math.LO↗

arXiv2017

Finding bases of uncountable free abelian groups is usually difficult

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.

math.LO↗