A New Approach to Bernoulli Convolutions and Salem Numbers

表现出某种自相似分形结构的集合和测度在自然界中非常常见，在纯数学中也经常出现。伯努利卷积也许是最简单的例子分形措施“与重叠”和问题的豪斯多夫维数的伯努利卷积是一个重要的数学问题，最初可以追溯到20世纪30年代。第一个例子伯努利卷积的维数小于一是由鄂尔多斯在1939年，不久后离开曼彻斯特大学。从那时起，许多数学学科的研究人员，包括数论，调和分析，加法组合学，分形几何和动力系统，都试图更好地理解伯努利卷积的维数理论。最近已经有了很大的进展，但一个完整的理论仍然难以捉摸。在这个项目中，我们试图结合联合收割机的申请人与秋山，冯和Ewesson在伯努利卷积的研究，使用一个家庭的矩阵与思想的Mercat，允许一个充分描述这些矩阵在特定情况下的最新进展。特别是，我们将花时间访问Mercat在马赛和结合我们的想法将表明，一个特定的伯努利卷积相关的塞勒姆数的维度。这是第一步证明著名的猜想，唯一的参数引起伯努利卷积的维数小于一个是皮索数，研究的例子鄂尔多斯在20世纪30年代。

Sets and measures exhibiting some kind of self-similar fractal structure are extremely common in nature, and come up surprisingly often in pure mathematics. Bernoulli convolutions are perhaps the simplest examples of fractal measures `with overlaps' and the question of the Hausdorff dimension of Bernoulli convolutions is an important mathematical problem, originally dating from the 1930s. The first examples of Bernoulli convolutions of dimension less than one were produced by Erdos in 1939, shortly after leaving the University of Manchester. Since then, researchers working across many mathematical disciplines, including number theory, harmonic analysis, additive combinatorics, fractal geometry and dynamical systems, have sought to understand better the dimension theory of Bernoulli convolutions. There has been very substantial recent progress but a complete theory remains elusive.In this project we seek to combine recent progress of the applicant together with Akiyama, Feng and Persson on the study of Bernoulli convolutions using a family of matrices with ideas of Mercat that allow one to fully describe these matrices in particular cases. In particular, we will spend time visiting Mercat in Marseille and by combining our ideas will show that a particular Bernoulli convolution associated to a Salem number has dimension one. This is a first step towards proving the famous conjecture that the only parameters giving rise to Bernoulli convolutions of dimension less than one are Pisot numbers, the examples studied by Erdos in the 1930s.