A Remark on Sets Having the Steinhaus Property

Boundedness and Surjectivity in Normed Spaces

We define the (w∗-) boundedness property and the (w∗-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w∗-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of tw...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

COUNTABLE COMPACTNESS AND THE LINDEL¨OF PROPERTY OF L-FUZZY SETS

In this paper, countable compactness and the Lindel¨of propertyare defined for L-fuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact L-set is countably compact and has the Lindel¨ofproperty. An L-set having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

A New Estimate for a Problem of Steinhaus

Steinhaus asked whether there exists a subset of the plane which, no matter how translated and rotated, always contains exactly one point with integer coordinates. It is still unknown if there exist sets with the Steinhaus property. Using harmonic analysis we prove that if a measurable set S R 2 satisses R S jxj 10 3 + dx < 1, for some > 0, then it cannot have the Steinhaus property. 0. Introdu...

On the Steinhaus tiling problem

We prove several results related to a question of Steinhaus: is there a set E ⊂ R such that the image of E under each rigid motion of R contains exactly one lattice point? Assuming measurability we answer the analogous question in higher dimensions in the negative, and we improve on the known partial results in the two dimensional case. We also consider a related problem involving finite sets o...