Restriction and Kakeya phenomena for finite fields

By G. Mockenhaupt, Terence Tao. 2002

151 citations recorded by the source corpus.

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these problems on finite fields. The restriction problem then becomes a question about general bounds for certain types of exponential sums, while the Kakeya problem has an algebraic geometry flavor, asking for the extent to which lines in different directions can overlap. In many cases the Euclidean arguments carry over easily to the finite setting (and are in fact somewhat cleaner), but there are some new phenomena in the finite case which deserve closer study.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e45afcffe58760784c7a2495301ded07c721b942

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In Applied Mathematics

1. Shubhangi Saraf
2. Madhu Sudan
3. Mark Lewko
4. Zeev Dvir
5. Frank de Zeeuw
6. Ben Green
7. Sophie Stevens
8. Larry Guth

In Mathematical Physics

1. Shubhangi Saraf
2. Zeev Dvir
3. Mark Lewko
4. Madhu Sudan
5. Ben Green

Citations in the corpus

• New Restriction Estimates for the 3-d Paraboloid over Finite Fields (2013, methodology): “In this section we will reprove the Stein-Tomas theorem from [14], and derive some related estimates which we need later.”
• Finite field models in additive combinatorics (2004, background): “We do not touch upon these matters here, referring the reader instead to the article [42] as well as in the surveys [37, 58, 59].”
• On the size of Kakeya sets in finite fields (2008, background): “bound was further improved in [Rog01, BKT04, MT04 , Tao08] both for general n and for specific small values of n (e.g for n = 3,4).”
• Improved lower bound on the size of Kakeya sets over finite fields (2008, background): “A bound of β ≤ 1/ √ 2 follows from [Mockenhaupt and Tao 2004] and the fact that products of Kakeya sets are Kakeya sets (in higher dimension).”
• An improved point‐line incidence bound over arbitrary fields (2016, background): “A contextual discussion of this problem as well as definitions of forthcoming notation are found in [24, 27].”