7 research outputs found
Topics in additive combinatorics and higher order Fourier analysis
Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-09-202
Quantitative bounds in the inverse theorem for the Gowers -norms over cyclic groups
We provide a new proof of the inverse theorem for the Gowers -norm
over groups for prime. This proof gives reasonable
quantitative bounds (the worst parameters are double-exponential), and in
particular does not make use of regularity or non-standard analysis, both of
which are new for in this setting.Comment: 123 page
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Expanders with Symmetry: Constructions and Applications
Expanders are sparse yet well-connected graphs with numerous theoretical and practical uses. Symmetry is a valuable structure for expanders as it enables efficient algorithms and a richer set of applications. This thesis studies expanders with symmetry, giving new constructions and applications. We extend expander construction techniques to work with symmetry and give explicit constructions of expanders with varying quality of expansion and symmetries of various groups. In particular, we construct graphs with large Abelian group symmetries via the technique of \textit{graph lifts}. We also give a generic amplification procedure that converts a weak expander to an almost optimal one while preserving symmetries. This procedure is obtained by generalizing prior amplification techniques that work for Cayley graphs over Abelian groups to Cayley graphs over any finite group. In particular, we obtain almost-Ramanujan expanders over every non-abelian finite simple group. We then explore the utility of having both symmetry and expansion simultaneously. We obtain explicit quantum LDPC codes of almost linear distance and \textit{good} classical quasi-cyclic codes with varying circulant sizes using prior results and our constructions of graphs with Abelian symmetries. We show how our generic amplification machinery boosts various structured expander-like objects: \textit{quantum expanders}, \textit{dimension expanders}, and \textit{monotone expanders}. Finally, we prove a structural result about expanding Cayley graphs, showing that they satisfy a \enquote{degree-2} variant of the \textit{expander mixing lemma}. As an application of this, we give a randomness-efficient query algorithm for \textit{homomorphism testing} of unitary-valued functions on finite groups and a derandomized version of the celebrated Babai--Nikolov--Pyber (BNP) lemma