Bernstein sets with algebraic properties

In this paper we consider a notion of I-Luzin set which generalizes the classical notion of Luzin set and Sierpiński set on Euclidean spaces. We show that there is a translation invariant σ-ideal I with Borel base for which I-Luzin set can be I-measurable. If we additionally assume that I has the Smital property (or its weaker version) then I-Luzin sets are I-nonmeasurable. We give some constructions of I-Luzin sets involving additive structure of R n . Moreover, we show that if c is regular, L is a generalized Luzin set and S is a generalized Sierpiński set then the complex sum L + S belongs to Marczewski ideal s 0 .

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“…In particular, we will work with so-called completely I-nonmeasurable sets which were examined in e.g. [5,9,12,16].…”

Section: Introductionmentioning

confidence: 99%

Some properties ofI-Luzin sets

Michalski

Żeberski

2015

Topology and its Applications

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“…Proof. We can take B to be a certain Bernstein subset of T that can be found using methods from [16]. The paper [16] For this set we then have that (7) fails for all m, n ∈ N. Indeed, we have on one hand…”

Section: On Further Extensions Of the Main Resultsmentioning

confidence: 99%

“…We can take B to be a certain Bernstein subset of T that can be found using methods from [16]. The paper [16] provides constructions of Bernstein subsets of R with additional algebraic properties. Using Method 3.2 and Application 3.3 from [16],…”

Section: On Further Extensions Of the Main Resultsmentioning

confidence: 99%

“…The paper [16] provides constructions of Bernstein subsets of R with additional algebraic properties. Using Method 3.2 and Application 3.3 from [16],…”

Section: On Further Extensions Of the Main Resultsmentioning

confidence: 99%

“…we can construct a Bernstein subset B ⊂ R such that B − B = R, and such that B + Z = B (for the latter property, which is not included in [16] explicitly, we can first ensure that 1 ∈ B, and since B is a subgroup we obtain the desired property; we omit the details). Thus, we obtain a Bernstein set B ⊂ T with the property that B − B = T.…”

Section: On Further Extensions Of the Main Resultsmentioning

confidence: 99%

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A Plünnecke–Ruzsa inequality in compact abelian groups

2019

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The Plünnecke-Ruzsa inequality is a fundamental tool to control the growth of finite subsets of abelian groups under repeated addition and subtraction. Other tools to handle sumsets have gained applicability by being extended to more general subsets of more general groups. This motivates extending the Plünnecke-Ruzsa inequality, in particular to measurable subsets of compact abelian groups by replacing the cardinality with the Haar probability measure. This objective is related to the question of the stability of classes of Haar measurable sets under addition. In this direction the class of analytic sets is a natural one to work with. We prove a Plünnecke-Ruzsa inequality for K-analytic sets in general compact (Hausdorff) abelian groups. We also discuss further extensions, some of which raise questions of independent interest in descriptive topology.

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Cp(X) for Hattori spaces

Tovar Acosta

2024

Topology and its Applications

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Bernstein sets with algebraic properties

Cited by 7 publications

References 5 publications

Some properties ofI-Luzin sets

Some properties ofI-Luzin sets

A Plünnecke–Ruzsa inequality in compact abelian groups

Cp(X) for Hattori spaces

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