Objetivo 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. Ámbito científico natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics Palabras clave spectral gap random walks on groups local limit theorem Bernoulli convolution self-similar measure Furstenberg boundary measure generic polynomial irreducibility of polynomials Programa(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Tema(s) ERC-2018-STG - ERC Starting Grant Convocatoria de propuestas ERC-2018-STG Consulte otros proyectos de esta convocatoria Régimen de financiación ERC-STG - Starting Grant Institución de acogida THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Aportación neta de la UEn € 1 334 109,00 Dirección TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge Reino Unido Ver en el mapa Región East of England East Anglia Cambridgeshire CC Tipo de actividad Higher or Secondary Education Establishments Enlaces Contactar con la organización Opens in new window Sitio web Opens in new window Participación en los programas de I+D de la UE Opens in new window Red de colaboración de HORIZON Opens in new window Coste total € 1 334 109,00 Beneficiarios (1) Ordenar alfabéticamente Ordenar por aportación neta de la UE Ampliar todo Contraer todo THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Reino Unido Aportación neta de la UEn € 1 334 109,00 Dirección TRINITY LANE THE OLD SCHOOLS CB2 1TN Cambridge Ver en el mapa Región East of England East Anglia Cambridgeshire CC Tipo de actividad Higher or Secondary Education Establishments Enlaces Contactar con la organización Opens in new window Sitio web Opens in new window Participación en los programas de I+D de la UE Opens in new window Red de colaboración de HORIZON Opens in new window Coste total € 1 334 109,00