Abstract
In this chapter, we present a technique, due to Jin, of converting some theorems about sets of positive upper asymptotic density or Shnirelmann density (to be defined below) to analogous theorems about sets of positive Banach density. The novel idea is to use the ergodic theorem for hypercycles applied to characteristic functions of nonstandard extensions. Deviating from Jin’s original formulation, we present his technique using the more recent notion of finite embeddability of subsets of natural numbers due to Di Nasso.
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- 1.
Recall that by Proposition 4.5 we have that \(X\in \mathcal {U}_\alpha \oplus \mathcal {U}_\beta \Leftrightarrow \alpha +{ }^*\beta \in { }^{\ast \ast }B\).
- 2.
The property that every \(B\in \mathcal {V}\) is piecewise syndetic is equivalent to the property that \(\mathcal {V}\) belongs to the closure of the smallest ideal \(K(\beta \mathbb {N},\oplus )\) (see [72, Theorem 4.40]).
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Working in the Remote Realm. 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_11
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DOI: https://doi.org/10.1007/978-3-030-17956-4_11
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