Malabika Pramanik

The Favard length of a Borel set E ⊂ ℝᵈ is the average value of the one-dimensional Hausdorff measure (i.e. length) of the orthogonal projection of E onto one-dimensional linear subspaces. We denote this average projected length by Fav(E). Due to the work of Besicovitch and Federer, a set E ⊂ ℝᵈ having positive and finite length satisfies Fav(E) = 0 if and only if E intersects all Lipschitz graphs in a set of null length. Such sets are called (purely 1)-unrectifiable sets.A quantitative reformulation of this result is as follows. Let δ > 0, and suppose that Eδ ⊂ ℝᵈ : (0,1) → (0,∞) such that Fav(Eδ) ≤ ψE(δ) for all δ ∈ (0,1) and lim(δ→0⁺) ψE(δ) = 0. The Favard length problem asks: what is the optimal choice of function ψE?Since 2001, the Favard length problem has seen significant attention within the field of mathematics known as geometric measure theory. For a class of unrectifiable sets in the plane known as rational product Cantor sets, there are strong asymptotic upper bounds (such as those of F. Nazarov, Y. Peres and A. Volberg) which are close to known sharp lower bounds (due to P. Mattila). These upper bounds rely upon Fourier analytic estimates, as well as combinatorial estimates, which demonstrate an emerging connection between the Favard length problem and size bounds for polynomials with many cyclotomic polynomial divisors.This dissertation presents the author’s work on the Favard length problem, with the main result being that the known planar upper bound for rational product Cantor sets in ℝ² generalizes completely to a class of rational product Cantor sets in ℝᵈ for d ≥ 3. The connection between the Favard length problem and cyclotomic divisibility is also explored in more detail. In particular, an explicit connection between the Favard length problem and the author’s recent size bounds for mask polynomials with many cyclotomic divisors is established.

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Harmonic analysis studies functions and sets through their frequencies. In harmonic analysis, central questions such as the restriction, Kakeya, and Bochner-Riesz conjectures involve oscillatory integrals. Geometric measure theory is a field that analyzes the geometric properties of sets in Euclidean spaces through measures supported on them. This dissertation concerns two problems at the intersection of harmonic analysis and geometric measure theory, with the common theme of applying oscillatory integral methods. The first problem involves notions of the dimensions of sets. The dimension is a key concept in geometric measure theory, which aims to capture the size of a set. The problem under consideration uses two distinct notions, Hausdorff and Fourier dimensions, in the context of hypersurfaces. Any hypersurface in Rᵈ⁺¹ has a Hausdorff dimension of d. However, the Fourier dimension depends on finer geometric properties of the hypersurface, such as curvature. For example, the Fourier dimension of a hyperplane is 0, and that of a hypersurface with non-vanishing Gaussian curvature is d. In 2022, Fraser, Harris, and Kroon showed that the Euclidean light cone in RRᵈ⁺¹ has a Fourier dimension of d-1, which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. This thesis proves this conjecture for all constant rank hypersurfaces. The method involves substantial generalizations of their strategy.The second problem involves recognizing quadratic patterns in R. Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations, such as arithmetic progressions and vertices of triangles, under the largeness of sets. In a seminal work by Laba and Pramanik in 2009, the largeness of a set was quantified by two measure-theoretic criteria, one involving the mass on Euclidean balls and the other using the decay properties of the Fourier transform of the measure. Recently, Kuca, Orponen, Sahlsten, and also Bruce, Pramanik removed the Fourier decay condition and showed that arbitrary sets with large Hausdorff dimensions contain two-point non-linear patterns, such as (x,y) and (x+t,y+t²). This thesis explores the existence of a three-point non-linear pattern x,x+t,x+t² in sets of large Hausdorff dimensions.

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This thesis addresses two distinct problems in geometric measure theory, both involving point configurations. The first is concerned with sets of positive Lebesgue measure that avoid affine copies of infinite configurations. The second is concerned with the Fourier dimensions of sets and the configurations they contain. When considering positive Lebesgue measure, we construct a fractal set and show how this construction avoids fast-decaying sequences. Before proving this result, we present some other constructions and strategies to address avoidance in sets of positive Lebesgue measure. Regarding the Fourier dimension, we not only consider avoidance but provide quantitative results that directly connect the Fourier dimension of a set to the inclusion of certain algebraic configurations within the set. We first introduce a technique that was introduced by Yiyu Liang and Malabika Pramanik, which is then generalized and applied to achieve the main results.

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The finite field Kakeya conjecture concerns the size of subsets of ?ⁿ? that contain a line in each direction, and is a discrete analogue of a major open problem in harmonic analysis. In 2008, Dvir found an elegant solution to this conjecture using elementary properties of polynomials. His proof popularized the polynomial method, which has proved to be a powerful strategy to tackle problems in analysis and discrete mathematics.This dissertation concerns two main research problems emerging from these areas. In the first, we consider a variant of the Kakeya problem. Besicovitch-Rado-Kinney (BRK) sets in ℝⁿ contain a sphere of radius ?, for each ? > 0. It is known that such sets have dimension ? from the work of Kolasa and Wolff. We consider a discrete version of this problem. We define BRK-type sets in ?ⁿ?, and establish lower bounds on the size of such sets using techniques introduced by Dvir’s proof of the finite field Kakeya conjecture.For our second main research problem, we study connections between hyperplanes and generalized polynomials in (ℤ/?ᵏℤ)ⁿ. Let ?ⁿ be the linear span of characteristic functions of hyperplanes in (ℤ/?ᵏℤ)ⁿ. We establish new upper bounds on the dimension of ?ⁿ over ℤ/?ℤ, or equivalently, on the rank of point-hyperplane incidence matrices in (ℤ/?ᵏℤ)ⁿ over ℤ/?ℤ. Our proof is based on a variant of the polynomial method using binomial coefficients in ℤ/?ᵏℤ as generalized polynomials. We also establish additional necessary conditions for a function on (ℤ/?ᵏℤ)ⁿ to be an element of ?ⁿ.

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In this thesis, we study two topics in Euclidean harmonic analysis. Thefirst one is the configurations contained in fractal-like sets in the Euclideanspace. The other is decoupling for various geometric objects in the Euclideanspace.In the study of Euclidean configurations, we first discuss the background,address their subtleties and do a simple survey on this subject. Then weproceed to the proof of my main result, which demonstrates the topologicalproperty of a set containing a similar copy of sequences converging to zero.In the study of decoupling, we first formulate a general decoupling inequality and discuss some general upper and lower bound estimates Thenwe move on to decoupling for manifolds in Euclidean space, and in particular curves in the plane. We then state a classical result by Bourgain andDemeter and use it to prove a decoupling inequality that works uniformlyfor all polynomials up to a certain degree, generalising an earlier result ofBiswas et al. in the plane.

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We discuss four different problems. The first, a joint work with Malabika Pramanik, concerns large subsets of ℝn that do not contain various types of configurations. We show that a collection of v points satisfying a continuously differentiable v-variate equation in ℝ can be avoided by a set of Hausdorff dimension 1/(v-1) and Minkowski dimension 1. The second problem concerns large subsets of vector spaces over non-archimedean local fields that do not contain configurations. Results analogous to the real-variable cases are obtained in this setting. The third problem is the construction of measure-zero Besicovitch-type sets in Kn for non-archimedean local fields K. This construction is based on a Euclidean construction of Wisewell and an earlier construction of Sawyer. The fourth problem, a joint work with Kyle Hambrook, is the construction of an explicit Salem set in ℚp. This set is based on a Euclidean construction of Kaufman. Supplementary materials available at: http://hdl.handle.net/2429/69960

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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 p +∞ if U contains infinitely many directionsin Rn. This unboundedness result for ℋU is an immediate consequence ofa lower estimate for ||ℋU||_Lp(ℝn) → Lp(ℝn) that we prove if the cardinality ofU is finite. In this case, we show that ||ℋU||_Lp(ℝn) → Lp(ℝn) is bounded frombelow by the square root of √log(#U) up to a positive constant depending only on p and n,for any exponent p in the range 1 p +∞ and any n ≥ 2. These resultswere first proved by G. A. Karagulyan () in the case n = p = 2.

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The main focus of this document is the small ball inequality. The small ball inequality is a functional inequality concerning the lower bound of the supremum norm of a linear combination of Haar functions supported on dyadic rectangles of a fixed volume. The sharp lower bound in this inequality, as yet unproven, is of considerable interest due to the inequality's numerous applications. We prove the optimal lower bound in this inequality under mild assumptions on the coefficients of a linear combination of Haar functions, and further investigate the lower bounds under more general assumptions on the coefficients. We also obtain lower bounds of such linear combinations of Haar functions in alternative function spaces such as exponential Orlicz spaces.

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Given a Cantor-type subset Ω of a smooth curve in ℝ(d+1), we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in ℝ(d+1), and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz , . In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(ℝ(d+1)) for all 1 ≤ p ∞.

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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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