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Report Garcia 2015

Listing research in progress, papers published or in preprint form

 Below is the description of a result that we have obtained, which is part of a work in progress.  

Doubling measures in Rn are measures that give some control on the size of the balls of Rnand appear frequently in many areas of Analysis and Geometry. Sets in Rn can be divided into six classes according to its evaluations by doubling measures in Rn. For example, very fat sets in Rn are sets that have positive measure for any doubling measure in Rn.

In  [Fat and Thin sets for doubling measures in Euclidean space, W. Wang, S. Wen, Z.-Y. Wen, Annales Acad. Scien. Fenn. 38 (2013), 535 -- 546)] it is considered the relation between the mentioned above classes of sets in R and its Cartesian products. For example, it is easily seen that if the Cartesian product of n sets in R is a fat set in Rn, then each of the n ‘component ’ sets must be fat in R. It is not known if the opposite implication is true, but it in that paper it is shown that this implication holds whenever the component sets belong to a family of symmetric Cantor sets, and also it is asked if this result could be obtained for more general Cantor sets.   

We could extend the above result of Wang et al: we showed that for a huge class of Cantor sets which were defined in [On Cantor sets and doubling measures, M. Csörnyei and V. Suomala, Journal of Math. Anal. and App. 393 (2012) 680 – 691] and have no symmetry assumption, the Cartesian product of fat sets in this family is also a fat set. We will continue working with problems in this direction.

As part of my next activities I will give two talks. One on Wednesday, October 21 in the Fractal Seminar and the other on Friday, October 23 in the Analysis Seminar, both seminars are held in the Pure Mathematics Faculty of the University of Waterloo.

It is probably likely that in November the Professor Franklin Mendivil, from Acadia University, visits the Pure Mathematics Faculty. We are thinking in starting something then, most likely related to computing the exact packing measure of certain self-similar like constructions in the plane. It is also expected that I visit Professor Mendivil during April 2016. This is because I have to teach in Waterloo until that month.


        The major outcome of my research visit is the possibility to engage in research with Kathryn Hare and other people from the Analysis group, in particular with people attending to the Fractal Seminar.