Summary
We show that if a set B of positive integers has positive upper density, then its difference set D(B) has extremely rich combinatorial structure, both additively and multiplicatively. If on the other hand only the density of D(B) rather than B is assumed to be positive, one is not guaranteed any multiplicative structure at all and is guaranteed only a modest amount of additive structure.
The first and third authors acknowledge support received from the National Science Foundation (USA) via grants DMS-9103056 and DMS-9025025 respectively. They also thank the US-Israel Binational Science Foundation for travel support.
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Bergelson, V., Erdős, P., Hindman, N., Łuczak, T. (2013). Dense Difference Sets and Their Combinatorial Structure. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_10
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DOI: https://doi.org/10.1007/978-1-4614-7258-2_10
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