Arithmetic progressions and the primes - El Escorial lectures

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.

We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik's Theorem. This version is a revision of the published version. Most notably, the properness hypotheses have been removed from Theorems 2 and 3, and the nu...

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

This paper has been withdrawn

A famous theorem of Szemer\'edi 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 t...

We develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. We apply it to obtain a new $L$-function free proof of Linnik's problem of bounding the least prime $p$ such that $p\equiv a\pmod q$ (with the bound $p \ll q^{350}$) as well as a new $L$-function free proof that the interval $(x-x^{39/40}, x]$ contains primes for every large $x$. In a future work we will develop the sieve further and provide more applications.

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.

We show estimates for the distribution of $k$-free numbers in short intervals and arithmetic progressions. We argue that, at least in certain ranges, these estimates agree with a conjecture by H. L. Montgomery.

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 non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to $1...

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.