1,707 research outputs found
Szemer\'edi'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 Szemer\'edi theorem given by
Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.Comment: 11 page
On Sums of Sets of Primes with Positive Relative Density
In this paper we show that if is a subset of the primes with positive
relative density , then must have positive upper density
in
. Our argument applies the techniques developed by Green and
Green-Tao used to find arithmetic progressions in the primes, in combination
with a result on sums of subsets of the multiplicative subgroup of the integers
modulo .Comment: 21 pages, to appear in J. London Math. Soc., short remark added and
typos fixe
On Improving Roth's Theorem in the Primes
Let be a set of prime numbers containing
no non-trivial arithmetic progressions. Suppose that has relative density
, where denotes the number of primes in the set
. By modifying Helfgott and De Roton's work, we
improve their bound and show that Comment: 14 pages, to appear in Mathematik
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 SzemerĂ©di 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 SzemerĂ©di theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative SzemerĂ©di theorem for k-term arithmetic progressions in pseudorandom subsets of â€_N of density N^(âck). The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative SzemerĂ©di theorem
A Multidimensional Szemer\'edi Theorem in the primes
Let be a subset of positive relative upper density of \PP^d, the
-tuples of primes. We prove that contains an affine copy of any finite
set F\subs\Z^d, which provides a natural multi-dimensional extension of the
theorem of Green and Tao on the existence of long arithmetic progressions in
the primes. The proof uses the hypergraph approach by assigning a pseudo-random
weight system to the pattern on a -partite hypergraph; a novel feature
being that the hypergraph is no longer uniform with weights attached to lower
dimensional edges. Then, instead of using a transference principle, we proceed
by extending the proof of the so-called hypergraph removal lemma to our
settings, relying only on the linear forms condition of Green and Tao
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