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Problems and Results on Intersective Sets

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Combinatorial and Additive Number Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 101))

Abstract

By intersective set we mean a set H ⊂ Z having the property that it intersects the difference set A − A of any dense subset A of the integers. By analogy between the integers and the ring of polynomials over a finite field, the notion of intersective sets also makes sense in the latter setting. We give a survey of methods used to study intersective sets, known results and open problems in both settings.

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Notes

  1. 1.

    If A ⊂ Z , then the upper density of A is defined by \(\overline{d}(A) = \overline{\lim }_{N\rightarrow \infty }\frac{\sharp A\cap \{1,\ldots,N\}} {N}\).

  2. 2.

    Roth’s theorem says that a set of positive upper density must contain non-trivial three-term arithmetic progressions.

  3. 3.

    The term intersective was coined by Ruzsa.

  4. 4.

    This definition is motivated by a theorem of van der Corput, which says that the set Z + is van der Corput.

  5. 5.

    Szemerédi’s theorem says that any dense subset of positive density of Z must contain arbitrarily long progression, a generalization of Roth’s theorem.

  6. 6.

    There is also the more general notion of sets multiple recurrence. A set H is called k-recurrence if whenever μ(A) > 0, there exists h ∈ H such that \(\mu (A \cap T^{-h}A \cap \cdots \cap T^{-kh}A) > 0\).

  7. 7.

    To be precise, equivalence classes of functions.

  8. 8.

    Actually, it is also a of multiple recurrence.

  9. 9.

    The relative upper density of A with respect to X is defined by \(\overline{d}_{X}(A) = \overline{\lim }_{N\rightarrow \infty }\frac{\sharp A\cap \{1,\ldots,N\}} {\sharp X\cap \{1,\ldots,N\}}\).

  10. 10.

    If h(0) = 0, then this is a special case of a theorem of Tao and Ziegler [42], which states that the primes contain configurations a, a + P 1(d), …, a + P k (d) for some a, d ∈ Z with d ≠ 0, where P 1, …, P k are given polynomials without constant term, another application of transference principles.

  11. 11.

    These are known as amenable groups.

  12. 12.

    In the definitions, one often have to make a choice either to consider all polynomials or just monic ones, but this doesn’t seem to matter.

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Acknowledgements

I would like to thank Noga Alon for helpful discussions and for his kind permission to include Theorem 5 and its proof. During the preparation of this paper, I was a member of the Institute for Advanced Study, and I would like to express my gratitude to the Institute for their generous support and hospitality.

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  1. Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX, 78712, USA

    Thái Hoàng Lê

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Correspondence to Thái Hoàng Lê .

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  1. Department of Mathematics, Lehman College (CUNY), Bronx, New York, USA

    Melvyn B. Nathanson

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Lê, T.H. (2014). Problems and Results on Intersective Sets. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_9

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