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.

Please contact Izabella laba or Malabika Pramanik for questions concerning any of the assignments and papers.

  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]