Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any $n,k\in \mathbb{N}$, any finite set with fewer than n elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero kth power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of $\mathbb{N}$ and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of $\mathbb{N}$ of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

拉姆齐理论的结果产生了许多自然的加和乘性概念。在本文中，我们解释了这些概念的子集之间的关系$\mathbb{ñ}$以及更一般的环论结构。我们表明，乘性以三种方式产生加性，并给出了一系列例证，证明了这些结果的最优性。由于加性和乘性之间的联系，我们还给出了各种应用。例如，我们证明给定任何$ñ，ķ\in \mathbb{ñ}$因此，在足够大的有限域中元素少于n个的任何有限集都可以转换，以使其每个元素变为非零的k次幂。我们还证明了关于Diophantine近似的定理，该定理沿的乘性综合子集存在。$\mathbb{ñ}$ 以及一个定理，表明在某些乘法半子群中，正上Banach密度的子集 $\mathbb{ñ}$零密度的整数包含任意长的算术级数。在此过程中，我们开发了一大类可修改半群中上Banach密度的新特征，并通过度量理论和拓扑动力学明确了递归定理的统一性。反过来，这导致了Szemerédi和van der Waerden关于算术级数的经典定理的增强形式。