Arbitrarily Long Arithmetic Progressions of Primes on the Nose

Arithmetic Progressions in the Graphs of Slightly Curved Sequences

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^α$ for some $α>0$. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. Furthermore, we extend Szemerédi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence $\{\lfloor{n^a}\rfloor\}_{n\in A}$ contains arbitrarily long a…

Strings of special primes in arithmetic progressions

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor πn\rfloor$ and some of arithmetic density zero such as primes of the form $…

Arithmetic progressions of Carmichael numbers in a reduced residue class

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a product of three distinct prime numbers.

Arithmetic progressions in multiplicative groups of finite fields

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-κ}$ and $r$ an arbitrarily fixed positive integer. Assuming $κ=κ(r)>0$ and $p$ large enough, it is shown that any proportional subset $A\subset G$ contains non-trivial arithmetic progressions of length $r$. The main ingredient is the Szemerédi-Green-Tao theorem.

The large $k$-term progression-free sets in $\mathbb{Z}_q^n$

Let $k$ and $n$ be fixed positive integers. For each prime power $q\geqslant k\geqslant 3$, we show that any subset $A\subseteq \mathbb{Z}_q^n$ free of $k$-term arithmetic progressions has size $|A|\leqslant c_k(q)^n$ with a constant $c_k(q)$ that can be expressed explicitly in terms of $k$ and $q$. As a consequence, we can take $c_k(q)=0.8415q$ for sufficiently large $q$ and arbitrarily fixed $k\geq 3$.

Almost arithmetic progressions in the primes and other large sets

A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note I provide a straightforward argument demonstrating that the primes get arbitrarily close to arbitrarily long arithmetic progressions. The argument also applies to `large sets' in the sense of Erdős-Turán. The proof is short, completely self-contained, and aims to give a heuristic explanation of why the primes, and other large sets, possess arithmetic structure.

The primes contain arbitrarily long polynomial progressions

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \leq m \leq x^\eps$ such that $x+P_1(m), ..., x+P_k(m)$ are simultaneously prime. The arguments are based on those in Green and Tao, which treated the linear case $P_i = (i-1)\m$ and $\eps=1$; the main new features are a localization of the shift parameters (and the attendant Gowers norm obj…

The dichotomy between structure and randomness, arithmetic progressions, and the primes

A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent applicatio…

Long arithmetic progressions of primes

This is an article for a general mathematical audience on the author's work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. It is based on several one hour lectures, chiefly given at British universities.

Nonrepetitive sequences on arithmetic progressions

A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every $k\geqslant 1$ and every $c\geqslant 1$ there exist arbitrarily long sequences over at most $(1+\frac{1}{c})k+18k^{c/c+1}$ symbols whose subsequences indexed by arithmetic progressions with common differences from the set $\{1,2,...,k\}$ are nonrepetitive. This improves a p…

Small gaps between configurations of prime polynomials

We find arbitrarily large configurations of irreducible polynomials over finite fields that are separated by low degree polynomials. Our proof adapts an argument of Pintz from the integers, in which he combines the methods of Goldston-Pintz-Yıldırım and Green-Tao to find arbitrarily long arithmetic progressions of generalized twin primes.

Szemerédi's theorem in the primes

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to zero sufficiently slowly contains a 3-term term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of 3-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem gi…

Arithmetic patches, weak tangents, and dimension

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including: the presence (or lack of) arithmetic progressions (or patches in dimensions $\geq 2$); the structure of tangent sets; and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dim…

A relative Szemerédi theorem

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 Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for $k$-term arithmetic progressions in…

The Green-Tao theorem for primes of the form $x^2+y^2+1$

We prove that the primes of the form $x^2+y^2+1$ contain arbitrarily long non-trivial arithmetic progressions.

Towards a sharp converse of Wall's theorem on arithmetic progressions

Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal for periodic increasing sequences $n_1<n_2<n_3<\dots$ of asymptotic density arbitrarily close to $1$, then $0.a_1a_2a_3\dots$ is normal. We show this is close to sharp in the sense that there are numbers $0.a_1a_2a_3\dots$ that are not normal, but for which $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal along a large collection of sequences w…

Sums of divisors on arithmetic progressions

For each $s\in \mathbb R$ and $n\in \mathbb N$, let $σ_s(n) = \sum_{d\mid n}d^s$. In this article, we give a comparison between $σ_s(an+b)$ and $σ_s(cn+d)$ where $a$, $b$, $c$, $d$, $s$ are fixed, the vectors $(a,b)$ and $(c,d)$ are linearly independent over $\mathbb Q$, and $n$ runs over all positive integers. For example, if $|s|\leq 1$, $a, b, c, d\in \mathbb N$ are fixed and satisfy certain natural conditions, then $$ σ_s(an+b) < σ_s(cn+d)\quad\text{ for all $n\leq M$} $$ where $M$ may be arbitrarily large, but in fact $σ_s(an+b) - σ_s(cn+d)$ has infinitely many sign changes. The results a…

The Green-Tao theorem: an exposition

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.

Obstructions to uniformity, and arithmetic patterns in the primes

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving ``randomly'', and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.

Constellations in prime elements of number fields

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consistin…

The Gaussian primes contain arbitrarily shaped constellations

We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, with $a \in \Z[i]$ and $r \in \Z \backslash \{0\}$, all of whose elements are Gaussian primes. The proof is modeled on a recent paper by Green and Tao and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the…

2-大集是零Bohr回归集

$2$-large sets are sets of nil-Bohr recurrence

本文证明2-大集与零Bohr邻域相交，解决了相关猜想与问题，还得出公差限制在零Bohr邻域时k项等差数列极值数的增长上限。