Izabella Laba

Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Mar 2019)
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 (2010-2017)
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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