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Equidistribution, fractal measures and arithmetic

Objective

The subject of this proposal lies at the crossroads of analysis, additive combinatorics, number theory and fractal geometry exploring equidistribution phenomena for random walks on groups and group actions and regularity properties of self-similar, self-affine and Furstenberg boundary measures and other kinds of stationary measures. Many of the problems I will study in this project are deeply linked with problems in number theory, such as bounds for the separation between algebraic numbers, Lehmer's conjecture and irreducibility of polynomials.

The central aim of the project is to gain insight into and eventually resolve problems in several main directions including the following. I will address the main challenges that remain in our understanding of the spectral gap of averaging operators on finite groups and Lie groups and I will study the applications of such estimates. I will build on the dramatic recent progress on a problem of Erdos from 1939 regarding Bernoulli convolutions. I will also investigate other families of fractal measures. I will examine the arithmetic properties (such as irreducibility and their Galois groups) of generic polynomials with bounded coefficients and in other related families of polynomials.

While these lines of research may seem unrelated, both the problems and the methods I propose to study them are deeply connected.

Call for proposal

ERC-2018-STG
See other projects for this call

Host institution

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
Address
Trinity Lane The Old Schools
CB2 1TN Cambridge
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 334 109

Beneficiaries (1)

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
United Kingdom
EU contribution
€ 1 334 109
Address
Trinity Lane The Old Schools
CB2 1TN Cambridge
Activity type
Higher or Secondary Education Establishments