EPSRC Reference: 
EP/T010835/1 
Title: 
A New Approach to Bernoulli Convolutions and Salem Numbers 
Principal Investigator: 
Kempton, Dr T 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Manchester, The 
Scheme: 
Overseas Travel Grants (OTGS) 
Starts: 
01 October 2019 
Ends: 
30 September 2021 
Value (£): 
28,700

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Sets and measures exhibiting some kind of selfsimilar 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.

Key Findings 
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Impacts 
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Summary 

Date Materialised 


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Further Information: 

Organisation Website: 
http://www.man.ac.uk 