A monad measure space for logarithmic density

@article{Nasso2015AMM,
  title={A monad measure space for logarithmic density},
  author={Mauro Di Nasso and Isaac Goldbring and Renling Jin and Steven C. Leth and Martino Lupini and Karl Mahlburg},
  journal={Monatshefte f{\"u}r Mathematik},
  year={2015},
  volume={181},
  pages={577-599}
}
We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $$A\subseteq \mathbb {N}$$A⊆N has positive Banach logarithmic density, then A contains an approximate geometric progression of any length. We also prove that if $$A,B\subseteq \mathbb {N}$$A,B⊆N have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $$A\cdot B$$A·B are multiplicatively bounded, a multiplicative… 
3 Citations

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