Roy, Manmatha2026-11-102026-10-29213p.http://hdl.handle.net/10263/7997This thesis has been completed under the supervision of Prof. Subhamoy Maitra (ASU) & Dr. Arijit Ghosh (ACMU)Fourier analysis of Boolean functions is a fundamental tool with far-reaching applications in complexity theory, learning theory, and cryptography. In many natural settings, functions of interest exhibit Fourier sparsity: they can be well approximated using only a small number of significant Fourier coefficients. In this thesis, we study Fourier-sparse Boolean functions from the perspective of property testing, where the goal is to infer global structural properties of an unknown function using only a small number of queries to its truth table, and where the efficiency of an algorithm is measured by its query complexity. Originating in the context of PCP constructions and later evolving into an independent area, property testing provides a natural lens for understanding the intrinsic complexity of combinatorial objects. Our first contribution provides a refined analysis of a sketching-by-hashing framework that operates directly in the Fourier domain. While such an approach was previously explored by Gopalan et al. (2011), we significantly strengthen its theoretical guarantees by deriving sharper structural bounds for the hashing process. As a consequence, we obtain a query-efficient local list-correction algorithm for the Hadamard code. This tool yields several applications, including query-efficient testers for distribution-free Fourier sparsity, testing membership in subclasses of Fourier-sparse boolean functions, linear isomorphism testing with respect to a known Boolean function, and sumset size estimation for set functions. All of these problems are well studied in learning theory, cryptography, and additive combinatorics. Our second contribution introduces a sketching-by-hashing framework based on subspace restrictions, operating in the input domain of Boolean functions. By analyzing how functions behave under random affine subspace restrictions, we establish several structural results that lead to improved algorithms for testing Fourier sparsity. The same framework enables estimation of the heaviest Fourier coefficient up to small additive error, with query complexity depending only on the desired accuracy. These problems are relevant across several areas of computer science. A Fourier sparsity tester, for instance, can serve as a preprocessing step for learning algorithms over Fourier-sparse function classes, validating whether an unknown function belongs to the target hypothesis class; the ability to estimate the heaviest Fourier coefficient, on the other hand, can be leveraged to design distinguishers for symmetric-key ciphers. Taken together, our work strengthens the connection between sketching-by-hashing techniques and the property testing of Fourier-sparse Boolean functions, and advances the understanding of query complexity for learning and testing in the Fourier domain. We expect the algorithmic techniques developed here to find further applications across many areas of computer science.enFourier AnalysisProperty TestingAlgorithmsQuery CompThesis