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 15: Restriction estimates, arithmetic patterns in thin sets, universality.
Topics 68: Differentiation theorems.
Topics 912: Exponential polynomial estimates and their applications.
Please contact Izabella laba or Malabika Pramanik for questions
concerning any of the assignments and papers.

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), 353365.
If time allows, mention also L. Kaufman, "On the theorem of Jarnik and Besicovitch",
Acta Arith. 39 (1981), 265267.
[presenter: X. Chen]

Salem sets and restriction properties of Fourier
transforms
by G. Mockenhaupt
Geom. Funct. Anal. 10 (2000), 15791587.
Focus on the proof of the restriction estimate.
[presenter: E. Palsson]

Extensions of the SteinTomas theorem.
by J.G. Bak, A. Seeger
Math.
Res. Lett. 18 (2011), 767781.
[presenter: M. Carnovale]

Arithmetic progressions in sets of fractional
dimension.
by I. Laba, M. Pramanik
Geom. Funct. Anal. 19 (2009), 429456.
Focus on the proof of the main result; skip
Sections 4 and 68.
If time allows, mention also:
T. Keleti: "A 1dimensional subset of the reals that intersects each of
its translates in at most a single point", Real Anal. Exchange 24 (1998/99), 843844.
[presenter: G. Amirkhanyan]

On a problem of Erd\H{o}s on sequences and measurable.
sets
by K. Falconer
Proc. Amer. Math. Soc. 90 (1984), 7778, and
and
Infinite patterns that can be avoided by measure.
by M. Kolountzakis
Bull. London Math. Soc. 29 (1997), 415424.
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),
333344
[presenter: K. Taylor]

6.
Averages in the plane over convex curves and maximal
operators.
by J. Bourgain
J. Analyse Math. 47 (1986), 6985.
[presenter: J. Zahl]

Maximal operators and differentiation theorems for
sparse sets.
by I. Laba, M. Pramanik
Duke Math. J. 158 (2011), 347411.
Part I:
Cover the derivation of maximal estimates
from the transversality conditions in Section 4.
[presenter: A. Rice]

Maximal operators and differentiation theorems for
sparse sets.
by I. Laba, M. Pramanik
Duke Math. J. 158 (2011), 347411.
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]

Bounded orthogonal systems and the Lambda_p problem.
by J. Bourgain
Acta
Math. 162 (1989), 227245.
Part I: Sections 13
[presenter: Mark Lewko]

Bounded orthogonal systems and the Lambda_p problem.
by J. Bourgain
Acta
Math. 162 (1989), 227245.
Part II: Sections 45.
[presenter: S. Steinerberger]

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, 366; translation in St. Petersburg Math.
J. 5 (1994), no. 4, 663717.
Part I: Focus on Turan's lemma and its proof.
Section 1.11.4 This presentation should also discuss (as much as possible,
at the expense of Section 1.4 if necessary):
An observation on TuranNazarov inequality
by O. Friedland and Y Yomdin
arxiv: 1107.0039
[presenter: C. Marx]

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, 366; translation in St. Petersburg Math.
J. 5 (1994), no. 4, 663717.
Part II: applications to uncertainty principle type theorems (Chapter 2)
[presenter: M. Bateman]

Projecting the onedimensional Sierpinski gasket.
by R. Kenyon
Israel
Journal of Mathematics 97 (1997), 221238.
If time allows, one can also mention:
G. Swiatek, J.J.P. Veerman, "On a conjecture of Furstenberg", Israel J.
Math. 130 (2002), 145155.
[presenter: E. Kroc]

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 23 and focus on the trigonometric
polynomial estimates.
[presenter: K. Hambrook]

Wiener's closure of translates' problem and PiatetskiiShapiro's uniqueness phenomenon
by N. lev, A. Olevskii
Ann. Math (2) 174 (2011) pp 519541
[presenter: A. Lewko]

Idempotents of Fourier multiplier algebra
by V. Lebedev, A. Olevskii
Geom. Funct. Anal. 4 (1994) , 539544
and
Fourier "L^p"  multipliers with bounded powers.
by V. Lebedev, A. Olevskii
(Russian) Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) pp 129166;
translation in Izv. Math 70 (2006) n0. 3 , 549585
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]