相关论文: A bilinear version of Bogolyubov's theorem

The bilinear Bogolyubov argument for $\mathbb{F}_p^n$ states that if we start with a dense set $A \subseteq \mathbb{F}_p^n \times \mathbb{F}_p^n$ and carry out sufficiently many steps where we replace every row or every column of $A$ by the…

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of…

The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums…

The Bogolyubov-Ruzsa lemma, in particular the quantitative bounds obtained by Sanders, plays a central role in obtaining effective bounds for the inverse $U^3$ theorem for the Gowers norms. Recently, Gowers and Mili\'cevi\'c applied a…

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns…

We prove a polynomial Bogolyubov type lemma for the special linear group over finite fields. Specifically, we show that there exists an absolute constant $C>0,$ such that if $A$ is a density $\alpha$ subset of the special linear group, then…

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a…

Let $G$ be a compact abelian group and $\phi_1, \phi_2, \phi_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\phi_i$'s, we prove the existence of Bohr sets in the sumset $\phi_1(A) + \phi_2(A) + \phi_3(A)$,…

The inverse theory for Gowers uniformity norms is one of the central topics in additive combinatorics and one of the most important aspects of the theory is the question of bounds. In this paper, we prove a quasipolynomial inverse theorem…

We establish new bounds in the Bogolyubov-Ruzsa lemma, demonstrating that if A is a subset of a finite abelian group with density alpha, then 3A-3A contains a Bohr set of rank O(log^2 (2/alpha)) and radius Omega(log^{-2} (2/alpha)). The…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

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…

Let $\{(A_i,B_i)\}_{i=1}^{m}$ be a collection of pairs of sets with $|A_i|=a$ and $|B_i|=b$ for $1\leq i\leq m$. Suppose that $A_i\cap B_j=\emptyset$ if and only if $i=j$, then by the famous Bollob\'{a}s theorem, we have the size of this…

Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$…

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting $G$ be a countable discrete abelian group and $\phi_1, \phi_2, \phi_3: G \to G$ be commuting endomorphisms whose images have finite…

A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give…

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of…

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a…

Bochner's theorem characterizes positive definite functions on groups through the positivity of their Fourier transforms and plays a fundamental role in Harmonic analysis. While Bochner-type results are known for certain classes of…

We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers…