We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse \(2^\omega \) to \(\omega \), while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

中文翻译：

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Mathias 和按密度参数化的银强迫

我们定义和研究了关于低密度和高密度的 Silver 和 Mathias 强迫的版本。我们专注于适当性、公理 A、链条条件、基数的保存和添加 Cohen 实数。我们发现将\(2^\omega \)折叠为\(\omega \)的粗暴强迫，而其他强迫则出奇地温和。我们还研究了由这些参数化强迫概念引起的规律性属性与 Baire 属性之间的联系。