Malabika Pramanik

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Configurations in fractal sets in Euclidean and non-Archimedean local fields (2018)

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.

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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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The small ball inequality with restricted coefficients (2016)

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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Kakeya-type sets, lacunarity, and directional maximal operators in Euclidean space (2015)

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 [4], [3]. 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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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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