Izabella Laba

Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - April 2022)
The unboundedness of the Maximal Directional Hilbert Transform (2018)

In this dissertation we study the maximal directional Hilbert transform operatorassociated with a set U of directions in the n-dimensional Euclideanspace. This operator shall be denoted by ℋU. We discuss in detail theproof of the (p; p)-weak unboundedness of ℋU in all dimensions n ≥ 2 andall Lebesgue exponents 1
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Restriction Theorems and Salem Sets (2015)

In the first part of this thesis, I prove the sharpness of the exponent range in the L² Fourier restriction theorem due to Mockenhaupt and Mitsis (with endpoint estimate due to Bak and Seeger) for measures on ℝ. The proof is based on a random Cantor-type construction of Salem sets due to Laba and Pramanik. The key new idea is to embed in the Salem set a small deterministic Cantor set that disrupts the restriction estimate for the natural measure on the Salem set but does not disrupt the measure's Fourier decay. In the second part of this thesis, I prove a lower bound on the Fourier dimension of Ε(ℚ,ψ,θ) = {x ∊ ℝ : ‖qx - θ‖ ≤ ψ(q) for infinitely many q ∊ ℚ}, where ℚ is an infinite subset of ℤ, Ψ : ℤ → (0,∞), and θ ∊ ℝ. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. I also prove a multi-dimensional analog of this result. I give applications of these results to metrical Diophantine approximation and determine the Hausdorff dimension of Ε(ℚ,ψ,θ) in new cases.

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Finite configurations in sparse sets (2014)

We prove a result which adds to the study of continuous analogues of Szemerédi-type problems. Let E ⊆ ℝⁿ be a Lebesgue-null set of Hausdorff dimension α, k, m be integers satisfying a suitable relationship, and {B₁,…, Bk} be n × (m − n) matrices. We prove that if the set of matrices Bi are non-degenerate in a particular sense, α is sufficiently close to n, and if E supports a probability measure satisfying certain dimensionality and Fourier decay conditions, then E contains a k-point configuration of the form {x + B₁y,…,x + Bky}. In particular, geometric configurations such as collinear triples, triangles, and parallelograms are contained in sets satisfying the above conditions.

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Structure and Arithmetic in Sets (2011)

We prove results in arithmetic combinatorics involving sums of prime numbers and also some variants of the Erdös-Szemerédi sum-product phenomenon. In particular, we prove nontrivial lower bounds on the density in the integers of the sumset of a positive relative density subset of the primes. The proof of this result uses Green and Green-Tao pseudorandomness arguments to reduce the problem to an analogous statement for relatively dense subsets of the multiplicative subgroup of integers modulo a large integer N. The latter statement is resolved with a combinatorial argument which bounds high moments of a representation function. We also show that if two distinct sets A and B of complex numbers have very small productset, then they produce maximally large iterated sumsets. This uses an algebraic concept of the multiplicative dimension of a finite set. As an application of the case A=B, we obtain a quantitative version of a result of Chang on sums and products of distinct complex elements.

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Arithmetic Structures in Random Sets (2008)

No abstract available.

Master's Student Supervision (2010 - 2021)
A field guide for Hilbert transforms with new estimates on an associated maximal directional operator (2021)

We give an overview of Hilbert transforms, followed by new results concerning maximal directional Hilbert transforms. Historically, the Hilbert transform motivated the development of many tools in harmonic analysis, such as interpolation theorems and more general singular integrals. Over time, variants of the Hilbert transform were studied as prototypical examples of singular integrals and maximal directional operators. In our research, we are especially concerned with maximal directional Hilbert transforms. After rigorously constructing the Hilbert transform and directional Hilbert transforms, we proceed to define the maximal directional Hilbert transforms. We then prove general L² mapping estimates for maximal directional Hilbert transforms, followed by specific examples which sharpen these estimates. Finally, we prove sharp L²(R²) to L²(R²) estimates for a large class of maximal directional Hilbert transforms.

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Fractal uncertainty principles for ellipsephic sets (2021)

Fractal uncertainty principles (FUPs) in harmonic analysis quantify the extent to which a function and its Fourier transform can be simultaneously localized near a fractal set. We investigate the formulation of such principles for ellipsephic sets, discrete Cantor-like sets consisting of integers in a given base with digits in a specified alphabet. We employ a combination of theoretical and numerical methods to find and support our results.To wit, we resolve a conjecture of Dyatlov and Jin by constructing a sequence of base-alphabet pairs whose FUP exponents converge to the basic exponent and whose dimensions converge to δ for any given δ ∊ (½, 1), thereby confirming that the improvement over the basic exponent may be arbitrarily small for all δ ∊ (0, 1). Furthermore, using the theory of prolate matrices, we show that the exponents β₁ of the same sequence decay subexponentially in the base.In addition, we explore extensions of our work to higher-order ellipsephic sets using blocking strategies and tensor power approximations. We also discuss the connection between discrete spectral sets and base-alphabet pairs achieving the maximal FUP exponent.

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Kakeya maximal function conjecture for semialgebraic mappings (2021)

The Kakeya maximal function conjecture is a quantitative, single scale formulation of the Kakeya conjecture. Recently, algebraic methods have been leading to progress in the Kakeya family of problems. In 2018, Katz and Rogers proved a conjecture concerning the number of ?-tubes with ?-separated directions which intersect a semialgebraic set with proportion at least λ. We will discuss the proof of this result which involves real algebraic geometry. We will then use this result to prove the Kakeya maximal function conjecture for the special case when the mappings are semialgebraic.

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Multilinear restriction estimates on fractal sets (2019)

In harmonic analysis there is a rich history of restriction theory for measures supported on smooth manifolds, and recently much focus has turned to restriction for measures supported on fractal sets. On the other hand, the use of multilinear restriction estimates has propelled most current progress on classical restriction theory. In this thesis we discuss the existing literature on both of these main interests in restriction theory, and then consider their combination. We analyze the existence of multilinear restriction estimates for a collection of singular measures, particularly measures supported on Cantor sets. We generalize a linear restriction estimate of Chen to a multilinear setting and provide a class of Cantor sets to which this result applies. Furthermore, we give necessary conditions for the existence of multilinear restriction for singular measures. We are hopeful that the success of multilinear restriction estimates in furthering classical restriction theory may be reproduced in our context.

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The finite field restriction problem (2011)

This work studies the extension problem for subsets of finite fields. This remains an important unsolved problem in harmonic analysis, in both the Euclidean and finite field setting. We survey the partial results obtained to date, common techniques, and open conjectures. In the case of a homogeneous variety H over a d-dimensional finite field, the L² to L⁴ boundedness is proved whenever H contains no hyperplanes. This is accomplished by proving an incidence theorem for cones of this type, and applying a sufficient condition for L² to L²m obtained by Mockenhaupt and Tao in their 2004 introductory paper. We moreover present counterexamples for particular cones when Γ
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