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Furstenberg problem for higher dimensional flats over F_q, R, Z_p

Speaker: Manik Dhar (MIT)
Speaker: Manik Dhar (MIT)
When Jun 04, 2024
from 04:00 PM to 05:00 PM
Where LH-111, First Floor
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Abstract: A Kakeya set in R^n is a set that contains a line segment in every direction. The Kakeya conjecture states that these sets have dimension n (open for n>=3). Over F_q^n a Kakeya set is similarly defined as containing a line in every direction. A breakthrough by Zeev Dvir using the polynomial method showed that Kakeya sets must have size C_n q^n.
In this talk, we consider some works on a generalization of this problem for higher dimensional flats. Furstenberg sets in F_q^n, R^n, Z_p^n are sets that have large intersections with k flats in every direction. For k>= 2 in F_q^n, Z_p^n, and k >= log_2 n for R^n we will show that these sets are large and give a very simple description of all tight examples. These results over finite fields have recently had surprising applications in the study of lattice coverings and linear hash functions. Based on works with Zeev Dvir, Ben Lund, and upcoming work with Paige Bright.
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