195,751 research outputs found
Separation with restricted families of sets
Given a finite -element set , a family of subsets is said to separate if any two elements of are separated by at
least one member of . It is shown that if ,
then one can select members of that
separate . If for some , then
members of
are always sufficient to separate all pairs of elements of that are
separated by some member of . This result is generalized to
simultaneous separation in several sets. Analogous questions on separation by
families of bounded Vapnik-Chervonenkis dimension and separation of point sets
in by convex sets are also considered.Comment: 13 page
On restricted families of projections in R^3
We study projections onto non-degenerate one-dimensional families of lines
and planes in . Using the classical potential theoretic
approach of R. Kaufman, one can show that the Hausdorff dimension of at most
-dimensional sets is typically preserved under
one-dimensional families of projections onto lines. We improve the result by an
, proving that if , then the
packing dimension of the projections is almost surely at least . For projections onto planes, we obtain a similar bound, with the
threshold replaced by . In the special case of self-similar sets without rotations, we obtain a full Marstrand type
projection theorem for one-parameter families of projections onto lines. The
case of the result follows from recent work of M.
Hochman, but the part is new: with this assumption,
we prove that the projections have positive length almost surely.Comment: 33 pages. v2: small changes, including extended introduction and
additional references. To appear in Proc. London Math. So
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
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