A New Approach to Bernoulli Convolutions and Salem Numbers

具有某种自相似分形结构的集合和度量在自然界中非常常见，在纯数学中出人意料地经常出现。伯努利卷积可能是具有重叠的分形度的最简单的例子，而伯努利卷积的Hausdorff维度的问题是一个重要的数学问题，最初可以追溯到20世纪30年代。1939年，在离开曼彻斯特大学后不久，鄂尔多斯制造了第一批维度小于1的伯努利卷积。从那时起，研究人员跨越了许多数学学科，包括数论、调和分析、加法组合学、分形几何和动力系统，试图更好地理解伯努利卷积的维度理论。在这个项目中，我们试图将申请人与秋山、冯和佩尔松在研究Bernoulli卷积方面的最新进展结合起来，使用一族矩阵和Mercat的思想，使人们能够在特定情况下完全描述这些矩阵。特别是，我们将花时间访问马赛的Mercat，通过结合我们的想法将表明与Salem数相关的特定Bernoulli卷积具有一维。这是证明著名猜想的第一步，该猜想是唯一能产生小于1维的伯努利卷积的参数是Pisot数，这是鄂尔多斯在20世纪30年代研究的例子。

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.