arXiv subjects
Publications and source records attributed to Thodoris Karageorgos.
We consider some variants of the Gowers box norms, introduced by Hatami, and show their relevance in the context of sparse hypergraphs. Our main results are the following. Firstly, we prove a generalized von Neumann theorem for $L_p$ graphons. Secondly, we give natural examples of pseudorandom families, that is, sparse weighted uniform hypergraphs which satisfy relative versions of the counting and removal lemmas.
We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.
We prove an algorithmic regularity lemma for $L_p$ regular matrices $(1 < p \leq \infty),$ a class of sparse $\{0,1\}$ matrices which obey a natural pseudorandomness condition. This extends a result of Coja-Oghlan, Cooper and Frieze who treated the case of $L_{\infty}$ regular matrices. We also present applications of this result for tensors and MAX-CSP instances.
We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.