Finite Field Restriction Estimates

1. Finite Field Restriction Estimates Mark Lewko TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA

2. What is Fourier analysis good for? If is small for Quantifying pseudo-randomness with respect to linear objects (equations/subspaces/subgroups/etc).

3. Character sums

4. Character sums Weil (1948) Deligne (1974) Bourgain (2005) Lots of Applications: Distribution / Security of RSA Distribution of quadratic residues Extractor constructions Gaps between primes (Etc.)

5. Restriction estimates attempt to understand exponential sums with arbitrary coefficients What can we hope to say? For most is small. Estimate: in terms of:

6. Let us reformulate the goal: surface measure on Estimates get better (harder to prove) as pdecreases and qincreases.

7. Finite Field Restriction Conjecture for the Paraboloid Classify (p,q) such that holds with independent of the field size.

8. Finite Field Restriction Conjecture for the Paraboloid, II • Restriction Estimate: Extension Estimate:

9. Finite Field Restriction: Motivation • Model problem for Euclidean harmonic analysis Understanding Exponential sums with coefficients

10. The Fourier Transform The Fourier Restriction Problem: Given a surface with measure can we define: for a.e. ?

11. The Fourier Restriction Problem I (3-d Sphere) (3-d Paraboloid) is continuous arbitrary in What about ?

12. The Fourier Restriction Problem II We want an inequality of the form: This is equivalent to the `extension’ inequality:

13. Score Board (3-d Sphere/Paraboloid): (trivial) Wolff 1995 Stein 1968 Tao, Vargas, Vega 1998 Tomas 1975 Tao, Vargas 2000 Tao 2002 Stein/Sjolin 1975 Bourgain 1991 Bourgain, Guth 2010 * Bourgain, Guth 2010

14. Geometric Properties

15. Geometric Properties II Overlap is the Enemy!

16. How much overlap can tubes have? KakeyaMaximal Conjecture Restriction Conjecture Kakeya Maximal Conjecture

17. Kakeya Set Conjecture If a set contains a line in every direction, how small can its dimension be? E E Restriction Conjecture Kakeya set Conjecture Kakeya Maximal

18. 3-d Kakeya Set Score Board Drury 1983 Bourgain 1991 Wolff 1995 Tao, Katz, Laba 1999

19. Back to Finite Fields

20. So what is the 3-d finite field restriction conjecture: Stein-Tomas Stein-Tomas Mockenhaupt, Tao 2002 L 2013 Bennett, Carbery, Garrigos, and Wright / Lewko-L 2010 L 2013 * L 2013

21. The Stein-Tomas method (doesn’t care if -1 is a square) Want to prove: (Parseval) (via Gauss Sums)

22. The Stein-Tomas method, I If

23. The Stein-Tomas method, II

24. The Stein-Tomas method, III Consider:

25. The Stein-Tomas method, IV We have proven:

26. How did Mockenhaupt-Tao go beyond Stein-Tomas? (-1 not a square) Extension estimate Restriction estimate E

27. Es Ls Mockenhaupt-Tao points lines # incidences*

28. Detour: Sum-product Estimates geometric progression arithmetic progression Erdősand Szemerédi’s sum-product conjecture: Erdős and Szemerédi’s (1983) …. Solymosi (2008)

29. Sum-product estimates (finite fields) (* not `near’ a subfield) Bourgain, Katz, Tao (2002) Szemerédi-Trotter Incidence Problem (finite fields) points lines # incidences (Cauchy-Schwarz) # incidences* (Bourgain, Katz, Tao)

30. Beyond Mockenhaupt-Tao Ls Es A) The Stein-Tomas / Mockenhaupt-Tao method isn’t sharp. points lines B) Each slice contains the same number of points, and is far from being contained in a subfield. # incidences* (Bourgain, Katz, Tao) The finite field restriction conjecture holds for:

31. What happens if -1 is a square?

32. -1 is a square, what goes wrong with the Mockenhaupt-Tao argument? Increase this exponent Want to go beyond S-T: But you need to decrease this exponent (and M-T needs to use the 2 for Parseval)

33. Let’s run the Mockenhaupt-Tao argument even though it can’t work E If the slices of E do not concentrate on lines then one can get some improvement one can get more out of the Stein-Tomas method Unless Consistent with the known problematic case:

34. If E concentrates on a plane: We can then geometrically understand E It is here were we have to (and do) avoid methods Being more careful, we can handle sets contained in planes

35. Last Case: Every slice of E is a line but E isn’t contained in a small number of planes.

36. Planes correspond to 1-d Fourier coefficients of Only potential problem is if all the planes stack up …but this can’t happen since we have assumed that the slices (green lines) don’t lie in small number of planes!

37. Stein-Tomas does better Summary of cases 1. 2. Most vertical slices don’t concentrate on lines Mockenhaupt-Tao argument 3. E is contained in a small number of planes E Direct computation using geometry of paraboloid 4. Slices of E are contained in lines, but E isn’t contained in a small number of planes M-T still bottleneck Can do better with sum-product Geometric estimate for the BR operator

38. Finite Field Kakeya conjecture finite field is a Kakeya set if it contains a line in every direction Finite Field Kakeya conjecture (Wolff):

39. Finite Field Kakeya How big must E be? Wolff ~1995 (elementary combinatorics) Bourgain, Katz, Tao 2002 (sum-product estimates) Dvir 2008 (Polynomial method)

40. What’s the relation between finite field restriction and Kakeya? (3-d Euclidean Paraboloid) One can’t do this in a finite field! Kakeya and restriction thought to be less connected over finite fields.

41. They are connected. Restriction for hyperbolic paraboloid in 2n-1 dimensions implies n dimensional Kakeya In odd dimensions with -1 a square this is equivalent to the standard paraboloid.

42. Consider

43. If we had a 3-d Kakeya set

44. Thank You!