Related papers: On the Decay of the Fourier Transform and Three Te…

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence…

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq…

We prove a version of Szemeredi's regularity lemma for subsets of a typical random set in F_p^n. As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed.

Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in Z_N…

A function $f:\mathbb{Z}_n \to \mathbb{C}$ can be represented as a linear combination $f(x)=\sum_{\alpha \in \mathbb{Z}_n}\widehat{f}(\alpha) \chi_{\alpha,n}(x)$ where $\widehat{f}$ is the (discrete) Fourier transform of $f$. Clearly, the…

Given positive integers $n$ and $k$, a $k$-term semi-progression of scope $m$ is a sequence $(x_1,x_2,...,x_k)$ such that $x_{j+1} - x_j \in \{d,2d,\ldots,md\}, 1 \le j \le k-1$, for some positive integer $d$. Thus an arithmetic progression…

In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This…

Let A \subseteq [1,..,N]^2 be a set of cardinality at least N^2/(log log N)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d>0. This theorem is a two-dimensional generalization of…

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial…

In this note we connect Sobolev estimates in the context of polynomial averages e.g. \[ \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \] whenever some $f_i$ vanishes…

Let H stand for the set of homeomorphisms on [0,1]. We prove the following dichotomy for Borel subsets A of [0,1]: either there exists a homeomorphism f in H such that the image f(A) contains no 3-term arithmetic progressions; or, for every…

In this paper we give a very elementary proof that if A and B are subsets of {1,2,...,N}, each having at least 5N^{1 - (4(k-1))^{-1}} elements, then the sumset A+B has a k-term arithmetic progression.

A generalized polymorphism of a predicate $P \subseteq \{0,1\}^m$ is a tuple of functions $f_1,\dots,f_m\colon \{0,1\}^n \to \{0,1\}$ satisfying the following property: If $x^{(1)},\dots,x^{(m)} \in \{0,1\}^n$ are such that…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

Szemer\'edi's theorem implies that there are $2^{o(n)}$ subsets of $[n]$ which do not contain a $k$-term arithmetic progression. A sparse analogue of this statement was obtained by Balogh, Morris, and Samotij, using the hypergraph container…

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has…

Let $\{H_{n,m}\}_{n,m\in \mathbb{N}}$ be the two dimensional Haar system and $S_{n,m}f$ be the rectangular partial sums of its Fourier series with respect to some $f\in L^1([0,1)^2)$. Let $\mathcal{N}, \mathcal{M}\subset \mathbb{N}$ be two…

We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than $1$, and…