Discrete Models of the Kakeya Set Problem and the Furstenberg Set Problem in R^n

Ruixiang Zhang
Princeton University

The "Minkowski version" of the Kakeya set problem and the Furstenberg set problem can be viewed as incidence problems about tubes and balls. We will present a result which solves the "discrete model" of both problems in R^n where the tubes and balls are replaced by lines and points, respectively. This shows that counterexamples of the real problems cannot be constructed "in a discrete way". The ingredients of the proof include a combination of the polynomial method and a theorem about tubular neighborhoods of varieties which goes back to Wongkew. I will discuss the proof and relevant open problems if time permits.

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