Related papers: An arithmetic regularity lemma, associated countin…
In this paper we analyze the practical implications of Szemer\'edi's regularity lemma in the preservation of metric information contained in large graphs. To this end, we present a heuristic algorithm to find regular partitions. Our…
A method for compression of large graphs and non-negative matrices to a block structure is proposed. Szemer\'edi's regularity lemma is used as heuristic motivation of the significance of stochastic block models. Another ingredient of the…
Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that…
This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed…
Recently Conlon, Fox, and the author gave a new proof of a relative Szemer\'edi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions.…
We prove a polynomial bound in the "true complexity" problem of Gowers and Wolf. The proof uses only repeated applications of the Cauchy--Schwarz inequality, answering negatively a question posed by Gowers and Wolf. To choose and reason…
Green and Tao's arithmetic regularity lemma and counting lemma together apply to systems of linear forms which satisfy a particular algebraic criterion known as the `flag condition'. We give an arithmetic regularity lemma and counting lemma…
The well-known regularity lemma of E. Szemer\'edi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It…
Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing…
The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of…
The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…
We prove an effective version of the inverse theorem for the Gowers $U^3$-norm for functions supported on high-rank quadratic level sets in finite vector spaces. For configurations controlled by the $U^3$-norm (complexity-two…
This is a companion note to our paper 'A relative Szemer\'edi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemer\'edi theorem for $(r+1)$-term arithmetic progressions assuming a linear…
We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of…
A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…
This paper establishes lower bounds for two kinds of arithmetic regularity partitions, building on constructions of Green [arXiv:math/0310476v2] and Hosseini, Lovett, Moshkovitz, and Shapira [arXiv:1405.4409]. The first kind occurs in the…
The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform…
We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse…
We are living in a world which is getting more and more interconnected and, as physiological effect, the interaction between the entities produces more and more information. This high throughput generation calls for techniques able to…
Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f) of a function f: G -> \C, where G is a finite abelian group and \C are the complex numbers. Roughly speaking, if U^d(f) is small, then f has certain…