# SUMMER SCHOOL Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics

The school will focus on questions at the interface of harmonic analysis, geometric measure theory and additive combinatorics, including applications of harmonic analytic and additive combinatorial method (restriction theorems, trigonometric polynomial estimates) to measure theoretic problems. In particular, many questions in geometric measure theory and harmonic analysis (e.g. concerning projections of sets or the occurrence of prescribed patterns) explore various concepts of "randomness" of sets. Similar phenomena for discrete sets have been investigated in additive combinatorics, and we hope that the powerful methods developed there can be transferred to a continuous setting. The school will introduce the participants to a selection of problems of this type. A basic familiarity with measure theory, harmonic analysis and probability theory will be expected.

The list of topics below is organized as follows.
Topics 1-5: Restriction estimates, arithmetic patterns in thin sets, universality.
Topics 6-8: Differentiation theorems.
Topics 9-12: Exponential polynomial estimates and their applications.

1. On singular monotonic functions whose spectrum has a given Hausdorff dimension Growth and generation in SL2(Z/pZ).
by R. Salem
Ark. Mat. 1 (1950), 353--365.
If time allows, mention also L. Kaufman, "On the theorem of Jarnik and Besicovitch", Acta Arith. 39 (1981), 265--267.
[presenter: X. Chen]
2. Salem sets and restriction properties of Fourier transforms
by G. Mockenhaupt
Geom. Funct. Anal. 10 (2000), 1579--1587.
Focus on the proof of the restriction estimate.
[presenter: E. Palsson]
3. Extensions of the Stein-Tomas theorem.
by J.-G. Bak, A. Seeger
Math. Res. Lett. 18 (2011), 767--781.
[presenter: M. Carnovale]
4. Arithmetic progressions in sets of fractional dimension.
by I. Laba, M. Pramanik
Geom. Funct. Anal. 19 (2009), 429-456.
Focus on the proof of the main result; skip Sections 4 and 6--8.

If time allows, mention also: T. Keleti: "A 1-dimensional subset of the reals that intersects each of its translates in at most a single point", Real Anal. Exchange 24 (1998/99), 843--844.
[presenter: G. Amirkhanyan]

5. On a problem of Erd\H{o}s on sequences and measurable. sets
by K. Falconer
Proc. Amer. Math. Soc. 90 (1984), 77--78, and
and
Infinite patterns that can be avoided by measure.
by M. Kolountzakis
Bull. London Math. Soc. 29 (1997), 415-424.
If time allows, the same presentation could also cover parts of: J. Bourgain, "Construction of sets of positive measure not containing an affine image of a given infinite structure", Israel J. Math. 60 (1987), 333--344
[presenter: K. Taylor]
6. 6. Averages in the plane over convex curves and maximal operators.
by J. Bourgain
J. Analyse Math. 47 (1986), 69--85.
[presenter: J. Zahl]
7. Maximal operators and differentiation theorems for sparse sets.
by I. Laba, M. Pramanik
Duke Math. J. 158 (2011), 347-411.

Part I: Cover the derivation of maximal estimates from the transversality conditions in Section 4.
[presenter: A. Rice]
8. Maximal operators and differentiation theorems for sparse sets.
by I. Laba, M. Pramanik
Duke Math. J. 158 (2011), 347-411.

Part II: Cover the proof of the transversality conditions for the random construction in Section 5. Please contact Prof Laba or Pramanik for more details.
[presenter: P. Shmerkin]
9. Bounded orthogonal systems and the Lambda_p problem.
by J. Bourgain
Acta Math. 162 (1989), 227-245.
Part I: Sections 1-3
[presenter: Mark Lewko]
10. Bounded orthogonal systems and the Lambda_p problem.
by J. Bourgain
Acta Math. 162 (1989), 227-245.
Part II: Sections 4-5.
[presenter: S. Steinerberger]
11. Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type
by F. Nazarov
(Russian) Algebra i Analiz 5 (1993), no. 4, 3--66; translation in St. Petersburg Math. J. 5 (1994), no. 4, 663-717.
Part I: Focus on Turan's lemma and its proof. Section 1.1-1.4 This presentation should also discuss (as much as possible, at the expense of Section 1.4 if necessary): An observation on Turan-Nazarov inequality
by O. Friedland and Y Yomdin
arxiv: 1107.0039
[presenter: C. Marx]
12. Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type
by F. Nazarov
(Russian) Algebra i Analiz 5 (1993), no. 4, 3--66; translation in St. Petersburg Math. J. 5 (1994), no. 4, 663-717.
Part II: applications to uncertainty principle type theorems (Chapter 2)
[presenter: M. Bateman]
13. Projecting the one-dimensional Sierpinski gasket.
by R. Kenyon
Israel Journal of Mathematics 97 (1997), 221-238.
If time allows, one can also mention: G. Swiatek, J.J.P. Veerman, "On a conjecture of Furstenberg", Israel J. Math. 130 (2002), 145--155.
[presenter: E. Kroc]
14. Buffon's needle estimates for rational product Cantor sets.
by M. Bond, I. Laba, A. Volberg
http://arxiv.org/abs/1109.1031
Skip the reductions in Sections 2-3 and focus on the trigonometric polynomial estimates.
[presenter: K. Hambrook]
15. Wiener's closure of translates' problem and Piatetskii-Shapiro's uniqueness phenomenon
by N. lev, A. Olevskii
Ann. Math (2) 174 (2011) pp 519--541
[presenter: A. Lewko]
16. Idempotents of Fourier multiplier algebra
by V. Lebedev, A. Olevskii
Geom. Funct. Anal. 4 (1994) , 539--544
and
Fourier "L^p" - multipliers with bounded powers.
by V. Lebedev, A. Olevskii
(Russian) Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) pp 129--166; translation in Izv. Math 70 (2006) n0. 3 , 549--585
The second paper generalizes the result of the first one. The presenter will discuss the first ppaer in detail and give an overview of the extensions in the second paper.
[presenter: V. Chan]