Goldstern's Principle with respect to Hausdorff Measures

2025/12/28 by Tatsuya Goto, Goto, Tatsuya
Mathematics · #Mathematical Dynamics and Fractals #Limits and Structures in Graph Theory #Advanced Banach Space Theory

paper · doi:10.48550/arxiv.2512.22852

Abstract

This paper is a continuation of the paper [Got25] and studies Goldstern's principle, a principle about unions of continuum many null sets, further. The main result is that the Hausdorff measure version of Goldstern's principle for \boldsymbolΠ¹₁ sets fails in L, despite the fact that the Lebesgue measure version is true. Moreover, we show that this version holds provided that the measurable cardinal exists. Other various results regarding Goldstern's principle are established.

Citations

• Goldstern's principle about unions of null sets
• Algorithmic information, plane Kakeya sets, and conditional dimension
• An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

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• A general principle for Hausdorff measure
• Hausdorff Measure: Lost in Translation
• On Goldfeld's Proof of Siegel's Theorem
• Schmidt's theorem, Hausdorff measures and Slicing