CANTOR SETS, PROJECTIONS AND LACUNARY SEQUENCES

Speaker: Yuval Peres, UC Berkeley

Let K be a nonrectifiable self-similar set of Hausdorff
dimension 1, e.g. Garnett's "four corner set".
By Besicovich (1938) almost all projections of K
have zero length, but what is the average length of projections
of an epsilon-neighborhood of K? The best available quantitative
upper and lower bounds (obtained jointly with B. Solomyak) differ
sharply, and surprisingly involve the function log-star(x),
the height of a tower of iterated exponentials that first exceeds x.

The second problem I'll discuss appears purely combinatorial,
but also leads to examining certain Cantor-type sets.
Let {n_k} be a lacunary sequence, i.e. the ratio of
successive elements of the sequence is at least some q>1.
In 1987, Erdos asked for the chromatic number of a graph G
on the integers, where two integers are connected by an edge
iff their difference is in the sequence {n_k}.
Y. Katznelson found a connection to a Diophantine approximation
problem. In joint work with W. Schlag, we improve Katznelson's
bounds for both problems using the Lovasz local lemma.











On Tue, 11 Feb 2003, Terence Tao wrote:

> Dear Yuval,
>
> I was thinking about making arrangements for your visit to UCLA on Feb 28.
> Do you have definite arrival and departure times for your visit? I
> can make hotel arrangements at this point.  Also, if you have a title
> prepared as well that would be excellent.
>
> Hope to see you soon,
>
> Terry
>