A Kakeya set is defined as a compact subset of R^n which contains a line segment of length 1 in every direction. The Kakeya conjecture states that every Kakeya set has Minkowski and Hausdorff dimension equal to n. One of the reasons this conjecture is important is that in 1971, Charles Fefferman used Kakeya sets to construct a counterexample to the ball multiplier conjecture in Fourier analysis.  Fefferman's work shows that if the Kakeya conjecture is false, other important conjectures in Fourier analysis, the Fourier restriction conjecture and the Bochner-Riesz conjecture, would be false as well.  The Kakeya conjecture is still open for n > 3.

 

In this talk, I will give a general introduction to the Kakeya problem and describe an improved Hausdorff dimension estimate for Kakeya sets in R^5 and higher that display a special structural property called "stickiness."

 

Speaker: Neeraja Kulkarni, University of Rochester