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October 16, 2025Open Access

Splitting Families, Reaping Families, and Families of Permutations Associated with Asymptotic Density

Key Points

The study reveals that the cardinal characteristic s0 is equal to covM, providing insights into continuum interactions.
It is shown that r0 is equivalent to nonM, highlighting underlying relations within cardinal invariants.
The relationship dd{r, all} equals dd{1/2, all} for any r in (0,1), pointing to uniformity in cardinal characteristics.
Proving Con(dd{(0,1), {0,1}}^{rel} < non(N)) and Con(dd{all, all}^{rel} < non(N)) emphasizes foundational properties in cardinal characteristics.

Abstract

We investigate several relations between cardinal characteristics of the continuum related with the asymptotic density of the natural numbers and some known cardinal invariants. Specifically, we study the cardinals of the form sX, rX and ddₗ, ₘ introduced in arXiv: 2304. 09698 and arXiv: 2410. 21102, answering some questions raised in these papers. In particular, we prove that s₀= cov (M) and r₀= non (M). We also show that dd\ₑ\, all=dd\₁/₂\, all for all r (0, 1), and we provide a proof of Con (dd (₀, ₁), \₀, ₁\^rel< non (N) ) and Con (dd₀₋₋, ₀₋₋^rel< non (N) ).

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Cite This Study

David Valderrama (Thu,) studied this question.

synapsesocial.com/papers/68f04acce559138a1a06e706 https://doi.org/https://doi.org/10.48550/arxiv.2506.21059

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