213 research outputs found
Small union with large set of centers
Let be a fixed set. By a scaled copy of around
we mean a set of the form for some .
In this survey paper we study results about the following type of problems:
How small can a set be if it contains a scaled copy of around every point
of a set of given size? We will consider the cases when is circle or sphere
centered at the origin, Cantor set in , the boundary of a square
centered at the origin, or more generally the -skeleton () of an
-dimensional cube centered at the origin or the -skeleton of a more
general polytope of .
We also study the case when we allow not only scaled copies but also scaled
and rotated copies and also the case when we allow only rotated copies
A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets
In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff
dimension greater than 1, then its one-dimensional projection has positive
Lebesgue measure for almost-all directions. In this article, we give a
combinatorial proof of this theorem when K is the product of regular Cantor
sets of class C^{1+a}, a>0, for which the sum of their Hausdorff dimension is
greater than 1.Comment: 9 pages, 1 figure, referee suggestions incorporated, to appear in
Expositiones Mathematica
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