# On the interplay between notions of additive and multiplicative largeness and its combinatorial applications

@article{Bergelson2016OnTI, title={On the interplay between notions of additive and multiplicative largeness and its combinatorial applications}, author={Vitaly Bergelson and Daniel Glasscock}, journal={arXiv: Combinatorics}, year={2016} }

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and…

## 3 Citations

Geometric progressions in syndetic sets

- MathematicsArchiv der Mathematik
- 2019

In order to investigate multiplicative structures in additively large sets, Beiglböck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded…

Multiplicative combinatorial properties of return time sets in minimal dynamical systems

- MathematicsDiscrete & Continuous Dynamical Systems - A
- 2019

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those…

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