Upper Banach density ( $d^{*} (S)$ )

For

S \subseteq Z

, its upper Banach density is

d^{*} (S) = lim sup_{N \to \infty} sup_{m \in Z} \frac{∣ S \cap [ m , m + N - 1 ] ∣}{N} .

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- Furstenberg correspondence principle Upper Banach density

Furstenberg correspondence principle

The Furstenberg correspondence principle turns a subset

S \subseteq Z

of positive upper Banach density into a shift-invariant probability system and a positive-measure cylinder

A

such that recurrence times of

A

produce differences of elements of

S

.

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Furstenberg correspondence principle

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