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Covering the real line with translates of a zero-dimensional compact set

András Máthé (2011)

Fundamenta Mathematicae

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

Covering theorems for uniqueness and extended uniqueness sets

Alexander S. Kechris, Alain Louveau (1990)

Colloquium Mathematicae

Criterios de convergencia de medidas de conjuntos Riemann integrables.

Pedro Jiménez Guerra (1980)

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

Curvature bounds for neighborhoods of self-similar sets

Steffen Winter (2011)

Commentationes Mathematicae Universitatis Carolinae

In some recent work, fractal curvatures $C_{k}^{f} (F)$ and fractal curvature measures $C_{k}^{f} (F, \cdot)$ , $k = 0, ..., d$ , have been determined for all self-similar sets $F$ in $ℝ^{d}$ , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations...

Curvature integral and Lipschitz parametrization in 1-regular metric spaces.

Hahlomaa, Immo (2007)

Annales Academiae Scientiarum Fennicae. Mathematica

Curvature measures and fractals [Book]

Steffen Winter (2008)