Related papers: A Roth type theorem for dense subsets of $\mathbb{…

We prove that if $A$ is any set of prime numbers satisfying \[ \sum_{a\in A}\frac{1}{a}=\infty, \] then $A$ must contain a $3$-term arithmetic progression. This is accomplished by combining the transference principle with a density…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…

Suppose that $A \subset \mathbb{R}$ has positive upper density, \[ \limsup_{|I| \to \infty} \frac{|A \cap I|}{|I|} = \delta > 0,\] and $P(t) \in \mathbb{R}[t]$ is a polynomial with no constant or linear term, or more generally a non-flat…

We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs…

We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ of density at least $(\log\log{N})^{-c}$ contains a nontrivial progression of the form $x,x+y,x+y^2$. This is the first quantitatively effective version of the…

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of…

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many…

We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…

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…

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - \epsilon_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \}…

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…

We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's…

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having…

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1,…

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 say that the set of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$ is super smooth if $y=\log^KN$ for a large fixed constant $K$. We show that the Roth's theorem on arithmetic progressions is true in super smooth numbers case. This…

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert\ll N(\log\log N)^4/\log…

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $\epsilon > 0$ there is some $N_0(\epsilon)$ such that for every $N \ge…