In this paper we investigate three unsolved conjectures in geometriccombinatorics, namely Falconer's distance set conjecture, the dimension ofFurstenburg sets, and Erdos's ring conjecture. We formulate natural$\delta$-discretized versions of these conjectures and show that in a certainsense that these discretized versions are equivalent. In particular, it appearsthat to progress on any of these problems one must prove a quantitativestatement about the existence of sub-rings of $R$ of dimension 1/2.
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机译：在本文中，我们研究了几何组合学中的三个未解猜想，即Falconer距离集猜想，Furstenburg集的维数和Erdos环猜想。我们对这些猜想制定了自然的\\ delta $-离散化版本，并从某种意义上证明了这些离散化版本是等效的。特别地，似乎要在任何这些问题上取得进展，必须证明有关存在尺寸为1/2的$ R $子环的定量陈述。
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