# A proof of a sumset conjecture of Erd\H{o}s

@article{Moreira2018APO, title={A proof of a sumset conjecture of Erd\H\{o\}s}, author={Joel Moreira and Florian Karl Richter and D. Robertson}, journal={arXiv: Combinatorics}, year={2018} }

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.

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#### References

SHOWING 1-10 OF 52 REFERENCES

On a Sumset Conjecture of Erdős

- Mathematics
- Canadian Journal of Mathematics
- 2015

Abstract Erdős conjectured that for any set $A\,\subseteq \,\mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,\,C\,\subseteq \,\mathbb{N}$ such that $B\,+\,C\,\subseteq… Expand

SETS OF RECURRENCE OF Z-ACTIONS AND PROPERTIES OF SETS OF DIFFERENCES IN Z

- 2006

In 1936, P. Erdos and P. Turan conjectured that if a set A of positive integers satisfies the condition . . M ,. hm sup --> 0, n then A contains arbitrarily long arithmetic progressions. This… Expand

A note on sums of primes

- Mathematics
- 1990

Under the assumption of the prime /c-tuplets conjecture we show that it is possible to construct an infinite sequence of integers, such that the average of any two is prime. Recently Pomerance,… Expand

Sumsets Contained in Infinite Sets of Integers

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1980

Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition, and Erdos conjectured that there exist infinite sets B and C such that B + C ⊆ A. Expand

Ramsey Theoretic Consequences of Some New Results About Algebra in the Stone-Čech Compactification

- Mathematics
- 2005

Recently [4] we have obtained some new algebraic results about βN, the Stone-Cech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a… Expand

Sumset phenomenon in countable amenable groups

- Mathematics
- 2009

Jin proved that whenever A and B are sets of positive upper density in Z, A + B is piecewise syndetic. Jin’s theorem was subsequently generalized by Jin and Keisler to a certain family of abelian… Expand

Ultrafilters and combinatorial number theory

- Mathematics
- 1979

Our concern is with two areas of mathematics and a, possibly surprising, intimate connection between them. One is the branch of combinatorial number theory which deals with the ability, given a… Expand

Definable sets containing productsets in expansions of groups

- Mathematics
- 2017

Abstract We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show… Expand

On Density, Translates, and Pairwise Sums of Integers

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1982

If a question of Erdos about pairwise sums has a counterexample, then it has countereXamples with maximal density arbitrarily close to 1, and several results about translates of sets with positive-maximal density and with positive asymptotic-upper density are derived. Expand

A Survey of Problems in Combinatorial Number Theory

- Mathematics
- 1980

Publisher Summary This chapter presents a survey of problems in combinatorial number theory. The chapter discusses problems connected with Van der Waerden's and Szemeredi's theorem. Thus the chapter… Expand