相关论文: Subsets of F_p^n without three term arithmetic pro…

In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too…

For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq…

We show that subsets of $\mathbb{F}_q^{\infty}$ of large Fourier dimension must contain three-term arithmetic progressions. This contrasts with a construction of Shmerkin of a subset of $\mathbb{R}$ of Fourier dimension $1$ with no…

In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier…

Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least…

Inspired by the Croot-Lev-Pach breakthrough, Jordan Ellenberg and Dion Gijswijt have recently amazed the combinatorial world by proving that the largest size of a subset of $F_3^n$ with no 3-term arithmetic progressions is exponentially…

Bourgain and Chang recently showed that any subset of $\mathbb{F}_p$ of density $\gg p^{-1/15}$ contains a nontrivial progression $x,x+y,x+y^2$. We answer a question of theirs by proving that if $P_1,P_2\in\mathbb{Z}[y]$ are linearly…

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever…

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining…

An example is presented of a subset $A$ of $\mathbb Z_N$ of density $\alpha$ such that the largest non-trivial Fourier coefficient of the characteristic function of $A$ is very small, but the probability that a random arithmetic progression…

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such…

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$. For $q=3$, the problem of finding the largest subset…

We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…

We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a…

Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…

We provide an upper bound of the size of a subset A of F_p^n that does not admit a k-star of 3-APs (three-term arithmetic progressions). Namely, the subset A is assumed to contain no configuration of k 3-APs, sharing the middle term, such…