Department of Mathematical Sciences

850 West Dickson Street, Room 309

University of Arkansas

Fayetteville, AR 72701

P 479-575-3351

F 479-575-8630

E-mail: math@uark.edu

# 34th Spring Lecture Series

"New Developments on the Discrepancy Function and Related Results." A series of five
lectures by Michael Lacey (Georgia Institute of Technology).

April 15-17, 2009

### Participants

**Oleksandra Beznosova** (University of Missouri at Columbia)**Dmitriy Bilyk** (University of South Carolina)**Adam Bowers** (University of Missouri at Columbia)**Jeremy Chapman** (University of Missouri at Columbia)**David Covert** (University of Missouri at Columbia)**Valerie Granger** (University of Missouri at Columbia)**Philip T. Gressman** (University of Pennsylvania)**Mariah Hamel** (University of Georgia)**Phillip Harrington** (University of South Dakota)**Alex Iosevich** (University of Missouri at Columbia)**Nets Katz** (Indiana University)**Doowon Koh** (University of Missouri at Columbia)**Izabella Laba** (University of British Columbia, Canada)**Michael Lacey** (Georgia Institute of Technology)**Entao Liu** (University of South Carolina)**Neil Lyall** (University of Georgia)**Akos Magyar** (University of British Columbia, Canada)**Melissa Moe** (University of Missouri at Columbia)**Kabe Moen** (University of Kansas)**Konstantin Oskolkov** (University of South Carolina)**Katharine Ott** (University of Kentucky)**Marco Peloso** (Universita’ di Milano, Bicocca, Italy)**Malabika Pramanik** (University of British Columbia, Canada)**Steven Senger** (University of Missouri at Columbia)**Leonid Slavin** (University of Missouri at Columbia)**Alexander Stokolos** (De Paul University)**Betsy Stovall** (University of California at Berkeley)**Krystal Taylor** (University of Missouri at Columbia)**Vladimir Temlyakov** (University of South Carolina)**Mingrui Yang** (University of South Carolina)

### 2009 Organizers

TBA

The University of Arkansas Spring Lecture series are conferences organized every spring by the Department of Mathematical Sciences of the University of Arkansas. Each conference is focused on a specific topic chosen among the current leading research areas in Mathematics; a principal lecturer delivers a short, five-lecture course and selects a number of specialists who are invited to give talks on subjects closely related to the topic of the conference. Short talks by young Ph.D.s and finishing graduate students are solicited to complement the conference. Each Lecture Series has grown into an ideal opportunity for specialists and young researchers to meet and exchange ideas about topics at the forefront of modern mathematics.

National Science Foundation support for the Spring Lecture Series DMS0751330, March 15, 2008-February 28, 2010, $72345. Luca Capogna (Principal Investigator) and Loredana Lanzani (Co-Principal Investigator).

The Spring Lectures are usually sponsored by the NSF jointly with the University of Arkansas. The proceedings of several conferences have appeared in the series "University of Arkansas Lecture Notes in the Mathematical Sciences," published by John Wiley & Sons.

**Oleksandra Beznosova (University of Missouri)**

Title: Linear with respect to the $A_2$-constant of the weight $w$ bound on the norm
of the perfect dyadic operator on the weighted Lebesgue spaces $L^2(w)$.*Abstract: We prove a sharp version of the T(1) theorem for the the perfect dyadic
singular integral operator on the real line.*

**Dmitriy Bilyk (University of South Carolina)**

Title: Upper bounds in discrepancy theory*Abstract: I will talk about upper bounds in irregularities of distribution, i.e. point
sets which have low discrepancy in various senses. I will discuss some classical examples
as well as recent developments.*

**Philip Gressman (University of Pennsylvania)**

Title: Generalized moment-entropy equations on the real line.*Abstract: A well-known result in information theory states that, among probability
density functions on the real line with a given (finite) variance, the entropy is
maximized by a Gaussian. We will consider the related problem of estimating nonoscillatory
integrals on the real line from below in terms of quantities related to entropy.**The relatively elementary estimates obtained in this way have wide-ranging applications
in harmonic analysis, including an improvement of the work of Tao and Wright on 1-dimensional
averages.*

**Mariah Hamel (University of Georgia)**

Title: On Sums of Sets of Primes with Positive Relative Density*Abstract: We will show that if $A$ is a subset of the primes with positive relative
density, then $A+A$ must have comparable positive density in $\mathbb{Z}$. Our proof
combines Fourier analytic techniques of Green and Green-Tao with a combinatorial result
on sumsets of subsets of the multiplicative group of integers modulo $m$.**This is joint work with Karsten Chipeniuk.*

**Nets Katz (Indiana University)**

Title: Discrete analogs of the Kakeya problem*Abstract: Using the algebraic method of Dvir, we solve the Joints problem, proving
that any set of $N$ lines in $R^3$ describe at most $O(N^{{3 \over 2}})$ joints, where
a joint is a point at which three of the lines intersect in a noncoplanar fashion.
We solve a related problem of Bourgain.**Joint with L. Guth.*

**Alex Iosevich (University of British Columbia, Canada)**

Title: Finite point configurations in vector spaces over finite fields*Abstract: We shall see that if a subset of $F_q^d$ is sufficiently large, then it
contains a positive proportion of all finite point configurations up to congruence.*

**Izabella Laba (University of British Columbia)**

Title: The Favard length of product Cantor sets*Abstract: Let K_n be the n-th iteration of a self-similar Cantor set K in the plane.
The Favard length Fav(K_n) of K_n is defined as the average length of a 1-dimensional
projection of K_n. It is well known that if K has Hausdorff dimension 1, but is not
contained in a line, then Fav(K_n) goes to zero as n goes to infinity. The hard problem
is to estimate the exact rate of decay. Last year, Nazarov, Peres and Volberg obtained
a power-type upper bound for the 4-corner Cantor set.**In a joint work with Kelan Zhai, we extend their result to a more general class of
product Cantor sets.*

**Entao Liu (University of South Carolina)**

Title: Greedy is Good: Sparse Signal Recovery*Abstract: In the last few years, a new method called compressive sensing has changed
the conventional knowledge in acquisition and reconstruction of compressible signals.
In this talk, a new idea to weaken a greedy algorithm in compressive sensing will
be presented.*

**Neil Lyall (University of Georgia)**

Title: Polynomial Configurations in Difference Sets*Abstract: I plan to discuss some joint work with Akos Magyar concerning certain arithmetic
properties of dense sets of integer points.*

**Akos Magyar (University of British Columbia, Canada)**

Title: Singular Radon transforms on discrete nilpotent groups: a model case*Abstract: We study discrete analogues of singular averages associated to polynomial
curves on nilpotent groups, focusing on the special case of a cubic curve on the Heisenberg
group.**This is joint work with A. Ionescu, E.M. Stein and S. Wainger.*

**Kabe Moen (University of Kansas)**

Title: Sharp Weighted Bounds for Fractional Integral Operators*Abstract. The relationship between the operator norms of fractional integral operators
acting on weighted Lebesgue spaces and the constant associated to the weights. We
obtain analogous results for the fractional integral operator to Petermichl's sharp
weighted bounds for singular integral operators. We also obtain analogous results
for the fractional integral operator to some difficult open problems in weighted theory
for Calderon-Zygmund operators. Our results rely on a sharp o-diagonal version of
the extrapolation theorem of Rubio de Francia. As an application of our techniques
we obtain improved weighted Sobolev inequalities.*

**Kostya Oskolkov (University of South Carolina)**

Title: On "non-differentiable" function of B. Riemann, and Schroedinger equation*Abstract: In the talk, applications will be presented, of the Schroedinger equation
to the study of differential, oscillatory and multi-fractal properties of the functions
$$\mathcal R_\alpha=\sum_{n\in \mathbb Z\setminus \{0\}} \frac{e^{\pi i (tn^2+2xn)}{|n|^\alpha},
(t,x)\in \mathbb R^2$$. The imaginary part of $\mathcal R_2$ for x = 0 is $R = \sum_{n\in
\mathbb N} n^{-2} \sin \pi n^2 t}$. $R$ was proposed by B. Riemann as a plausible
example of a function that is continuous and nowhere differentiable.*

**Katharine Ott (University of Kentucky)**

Title: Spectral properties of the single layer potential operator for the Lame' system
of elastostatics on domains with isolated singularities*Abstract: We establish sharp well-posedness results for the regularity problem for
the Lam\'{e} system of elastostatics in the class of curvilinear polygons in two dimensions.
The key technical ingredient is obtaining spectral properties for the boundary version
of the single layer potential operator associated with the Lam\'{e} system acting
on $p$-integrable functions on the boundary. Our approach relies on Mellin transform
and global optimization techniques.**This is joint work with Irina Mitrea and Warwick Tucker.*

**Malabika Pramanik (University of British Columbia, Canada)**

Title: Maximal averages over sparse sets in $\mathbb R$*Abstract: We discuss $L^p$ mapping properties of a maximal operator associated with
averages against certain singular measures obtained by Cantor type constructions.
In particular, this answers a question of Aversa and Preiss concerning density theorems,
and extends a result of Rubio de Francia on a generalization of Stein's theorem for
the spherical maximal function. Comparisons and contrasts with Bourgain's circular
maximal theorem will be explored.**Joint work with Izabella Laba.*

**Alexander Stokolos (De Paul University)**

Title: A gentle introduction in the Bellman Function technique*Abstract: I will explain how the Bellman Function methodology is working using several
examples. In particular I will show how to find the Bellman Function for the dyadic
maximal operator (joint result with Leonid Slavin and Vasily Vasyunin).*

**Betsy Stovall (University of California at Berkeley)**

Title: Strong-type endpoint $L^p \to L^q$ estimates for certain generalized Radon
transforms*Abstract: A new technique of M. Christ allows one to prove strong-type $L^p \to L^q$
estimates using combinatorial methods which had previously only yielded restricted
weak-type bounds. We will discuss this technique and some applications of it.*

**Vladimir N. Temlyakov (University of South Carolina)**

Title: Cubature Formulas, Discrepancy, and Nonlinear Approximation*Abstract: The main goal of this talk is to demonstrate connections between the following
three big areas of research: the theory of cubature formulas (numerical integration),
the discrepancy theory, and nonlinear approximation. In particular, we will show how
standard in the theory of cubature formulas settings can be translated into the discrepancy
problem and into a natural generalization of the discrepancy problem. This leads to
a concept of the r-discrepancy. We also present results on a relation between construction
of an optimal cubature formula with m knots for a given function class and best nonlinear
m-term approximation of a special function determined by the function class. The nonlinear
m-term approximation is taken with regard to a redundant dictionary also determined
by the function class.*

**Mingrui Yang (University of South Carolina)**

Title: Greedy approximations with regard to multivariate bases with the tensor product
structure*Abstract: We compare the greedy algorithm with best $m$-term approximation on classes
with regard to various bases. More precisely, we determine the exact error bound for
both the greedy approximation and best $m$-term approximation on the class of functions
whose coefficients belong to some Lorentz-type sequence space, where the bases can
be taken as greedy bases, quasi-greedy bases or the tensor product of greedy bases.*

**SLS 2009 Archived Video Playlist**

**Michael Lacey (Georgia Institute of Technology)**

"New Developments on the Discrepancy Function and Related Results (Lecture 1)"

SLS 2009 - 01

**Dmitriy Bilyk (University of South Carolina)**

"Upper bounds in discrepancy theory"

SLS 2009 - 02

**Katharine Ott (University of Kentucky)**

"Spectral properties of the single layer potential operator for the Lame' system of
elastostatics on domains with isolated singularities"

SLS 2009 - 03

**Philip Gressman (University of Pennsylvania)**

"Generalized moment-entropy equations on the real line"

SLS 2009 - 04

**Michael Lacey (Georgia Institute of Technology)**

"New Developments on the Discrepancy Function and Related Results (Lecture 2)"

SLS 2009 - 05

**Betsy Stovall (UC Berkley)**

"Strong-type endpoint $L^p \to L^q$ estimates for certain generalized Radon transforms"

SLS 2009 - 06

**Kabe Moen (University of Kansas)**

"Sharp Weighted Bounds for Fractional Integral Operators"

SLS 2009 - 07

**W Larry Guth (Massachusetts Institute of Technology)**

"A point p ∈ IR^3 is joint of lines ℓ_1, ℓ_2, ℓ_3 if p ∈ ℓ_1, ℓ_2, ℓ_3 and ℓ_1, ℓ_2,
ℓ_3 are not coplanar"

SLS 2009 - 08

**Michael Lacey (Georgia Institute of Technology)**

"Mathematics of Futurama (Public Lecture)"

SLS 2009 - 09

**Michael Lacey (Georgia Institute of Technology)**

"New Developments on the Discrepancy Function and Related Results (Lecture 3)"

SLS 2009 - 10

**Izabella Laba (University of British Columbia)**

"The Favard length of product Cantor sets"

SLS 2009 - 11

**Mariah Hamel (University of Georgia)**

"On Sums of Sets of Primes with Positive Relative Density"

SLS 2009 - 12

**Alexander Stokolos (De Paul University)**

"A gentle introduction in the Bellman Function technique"

SLS 2009 - 13

**Michael Lacey (Georgia Institute of Technology)**

"New Developments on the Discrepancy Function and Related Results (Lecture 4)"

SLS 2009 - 14

**Oleksandra Beznosova (University of Missouri)**

"Linear with respect to the $A_2$-constant of the weight $w$ bound on the norm of
the perfect dyadic operator on the weighted Lebesgue spaces $L^2(w)$"

SLS 2009 - 15

**Entao Liu (University of South Carolina)**

"Greedy is Good: Sparse Signal Recovery"

SLS 2009 - 16

**Akos Magyar (University of British Columbia)**

"Singular Radon transforms on discrete nilpotent groups: a model case"

SLS 2009 - 17

**Alex Iosevich (University of Missouri)**

"Finite point configurations in vector spaces over finite fields"

SLS 2009 - 18

**Vladimir Temlyakov (University of South Carolina)**

"Cubature Formulas, Discrepancy, and Nonlinear Approximation"

SLS 2009 - 19

**Mingrui Yang (University of South Carolina)**

"Greedy approximations with regard to multivariate bases with the tensor product structure"

SLS 2009 - 20

**Konstantin Oskolkov (University of South Carolina)**

"On 'non-differentiable' function of B. Riemann, and Schroedinger equation"

SLS 2009 - 21

**Malabika Pramanik (University of British Colombia)**

"Maximal averages over sparse sets in $\mathbb R$"

SLS 2009 - 22

**Neil Lyall (University of Georgia)**

"Polynomial Configurations in Difference Sets"

SLS 2009 - 23

**Michael Lacey (Georgia Institute of Technology)**

"New Developments on the Discrepancy Function and Related Results (Lecture 5)"

SLS 2009 - 24