We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.
@article{bwmeta1.element.bwnjournal-article-fmv166i3p269bwm,
author = {Arnold Miller and Strashimir Popvassilev},
title = {Vitali sets and Hamel bases that are Marczewski measurable},
journal = {Fundamenta Mathematicae},
volume = {163},
year = {2000},
pages = {269-279},
zbl = {0968.03051},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv166i3p269bwm}
}
Miller, Arnold; Popvassilev, Strashimir. Vitali sets and Hamel bases that are Marczewski measurable. Fundamenta Mathematicae, Tome 163 (2000) pp. 269-279. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv166i3p269bwm/
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