Additive combinatorics of infinite sets via ergodic theoretic approach

拟议的项目将使用创新的遍历理论方法，使我们能够解决加法组合学(数论)和分数几何中的重要问题。特别是，我们将解决长期存在的无限集合的可加性逆问题，发现数论中的和积现象，并在分形集中发现过多的有限构形。我们还将把准晶体最流行的数学模型之一的结构理论推广到更广泛的基团类别。该项目将对加法组合学和遍历理论做出重大贡献，并将把澳大利亚在这些领域的研究推向更高的高度。

The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of one of the most popular mathematical models of quasi-crystals to a more extensive class of groups. This project will make significant contributions to Additive Combinatorics and Ergodic Theory and will bring the Australian research in these fields to ever greater heights.