Georgia Institute of Technology College of Sciences

We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain's sum-product theorem.

The first part, consists on a result in the area of commutators. The classic result by Coifman, Rochber and Weiss, stablishes a relation between a BMO function, and the commutator of such a function with the Hilbert transform. The result obtained for this thesis, is in the two parameters setting (with obvious generalizations to more than two parameters) in the case where the BMO function is matrix valued. The second part of the thesis corresponds to domination of operators by using a special class called sparse operators. These operators are positive and highly localized, and therefore, allows for a very efficient way of proving weighted and unweighted estimates. Three main results in this area will be presented: The first one, is a sparse version of the celebrated $T1$ theorem of David and Journé: under some conditions on the action of a Calderón-Zygmund operator $T$ over the indicator function of a cube, we have sparse control.. The second result, is an application of the sparse techniques to dominate a discrete oscillatory version of the Hilbert transform with a quadratic phase, for which the notion of sparse operator has to be extended to functions on the integers. The last resuilt, proves that the Bochner-Riesz multipliers satisfy a range of sparse bounds, we work with the ’single scale’ version of the Bochner-Riesz Conjecture directly, and use the ‘optimal’ unweighted estimates to derive the sparse bounds.

We propose a general phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks. The algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees are presented. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

Isospectral reductions decrease the dimension of the adjacency matrix while keeping all the eigenvalues. This is achieved by using rational functions in the entries of the reduced matrix. I will show how it's done through an example. I will also discuss about the eigenvectors and generalized eigenvectors before and after reductions.