Abstract
In this section, we discuss a conjecture of Erdős, which states that a set of natural numbers of positive lower density contains the sum of two infinite sets. We begin with the history of the conjecture and discuss its nonstandard reformulation. We then present a proof of the conjecture in the “high density” case, which follows from a “1-shift” version of the conjecture in the general case. We conclude with a discussion of how these techniques yield a weak density version of Folkman’s theorem.
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Notes
- 1.
It still seems to be open whether or not a set of positive density (of any kind) must contain a translate of \(\operatorname {PS}(B)\) for some infinite B.
- 2.
Indeed, if A is piecewise syndetic, then A + [0, k] is thick for some \(k\in \mathbb {N}\). Thick sets are easily seen to contain FS-sets, whence, by the Strong version of Hindman’s theorem (Corollary 8.7), A + i contains an FS-set for some i ∈ [0, k]. It follows immediately that A has the sumset property.
- 3.
Notice that at this point we have another proof of Nathanson’s Theorem 13.4: if we set B := {d n, d n+1, …} and C := {l 1, …, l n}, then B + C ⊆ A.
References
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P. Erdős, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6, 89–115 (1980)
P. Erdős, R.L. Graham, Old and New Problems and Results in Combinatorial Number Theory. Monographies de L’Enseignement Mathématique, vol. 28 (Université de Genève, L’Enseignement Mathématique, Geneva, 1980)
J. Moreira, F.K. Richter, D. Robertson, A proof of the Erdős sumset conjecture. arXiv:1803.00498 (2018)
M.B. Nathanson, Sumsets contained in infinite sets of integers. J. Comb. Theory Ser. A 28(2), 150–155 (1980)
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Sumset Configurations in Sets of Positive Density. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_13
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