The Group of Large Diffeomorphisms in General Relativity

Domenico Giulini

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The paper is concerned with the class of groups satisfying the finite embedding (FE) property. This is a generalization of residually finite groups. In [2] it was asked whether there exist FE-groups which are not residually finite. Here we present such examples. To do this, we construct a family of three-generator soluble FE-groups with torsion-free abelian factors. We study necessary and sufficient conditions for groups from this class to be residually finite. This answers the questions...

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The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an $L^{2}$ – $L^{p}$ restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime $k$ -tuples. Let $a_{1}, \dots, a_{k}$ and $b_{1}, \dots, b_{k}$ be positive integers. Write $h (θ) : = \sum_{n \in X} e (n θ)$ , where $X$ is the set of all $n \leq N$ such that the numbers $a_{1} n + b_{1}, \dots, a_{k} n + b_{k}$ are all prime. We obtain upper bounds for ${∥ h ∥}_{L^{p} (𝕋)}$ , $p > 2$ , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct...

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At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following : (1) Are there infinitely many primes of the form $n^{2} + 1$ ? ...