Description

We plan to study the following general principle:

The union of an m-parameter family (or more generally an m Hausdorff dimensional family) of k-dimensional planes of Rⁿ has positive Lebesgue measure if m+k>n and has Hausdorff dimension m+k otherwise.

Unfortunately, this is too good to be true, there exist very simple counter-examples. On the other hand, in some restricted cases (for example if k=n-1 or m is at most 1) this is known to be true, and more importantly, famous conjectures would (essentially) follow from some other restricted cases: The most famous is the surely extremely hard Kakeya conjecture (one of the favorite problems of Terry Tao), which states that any compact set in Rⁿ that has a unit line segment in every direction must have Hausdorff dimension n. An other example is the also famous and long standing conjecture that states that for any 1<k<n any Borel set in Rⁿ containing k-dimensional planes in every direction must have positive Lebesgue measure.

Of course, attacking these conjectures would be too ambitious. Instead, we have the following more realistic (but less well defined) goals:

1. to find and prove sufficient conditions that guarantee that the above principle holds,
2. to find and prove necessary conditions that are needed for the principle,
3. to find conjecture for the exact (necessary and sufficient) condition and
4. to find counter-examples for naive false conjectures.

A previous attempt to this problem in an earlier BSM Research Project and its continuation eventually led to the following joint paper with the students: https://arxiv.org/abs/1906.06288, which is to appear in Ann. Acad. Sci. Fenn. Math. This time we will try different directions.