Malabika Pramanik

Arithmetic Progressions in Sets of Fractional Dimension

Let

A Calderón–Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators

{E \subset\mathbb{R}}

Maximal operators and differentiation theorems for sparse sets

be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.

On polynomial configurations in fractal sets

We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions

A discrete carleson theorem along the primes with a restricted supremum

\geq 3

Large sets avoiding patterns

. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature. As an application, we define a new notion of the height of real-analytic functions, compute the critical integrability index, and obtain sharp growth rate of sublevel sets. This also leads to a characterization of the oscillation index of scalar oscillatory integrals with real-analytic phases in all dimensions.

A Roth type theorem for dense subsets of

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case.

\mathbb{R}^d

A two-dimensional weighted integral in R 2 is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.