# Séminaires : Séminaire Théorie des Nombres

 Equipe(s) : fa, tn, tga, Responsables : Marc Hindry, Bruno Kahn, Wieslawa Niziol, Cathy Swaenepoel Email des responsables : cathy.swaenepoel@imj-prg.fr Salle : Adresse : Description http://www.imj-prg.fr/tn/STN/stnj.html

 Orateur(s) Thomas Bloom - University of Oxford, Titre A density conjecture about unit fractions Date 28/03/2022 Horaire 14:00 à 15:00 Diffusion Résume In 2001 Croot resolved an old conjecture of Erdos and Graham, proving that in any finite colouring of the positive integers there is a (non-trivial) monochromatic solution to $\frac{1}{n_1}+\cdots+\frac{1}{n_k} = 1$ with all $n_i$ distinct. A natural generalisation, also conjectured by Erdos and Graham, is that in fact any set of positive density contains such a solution. We will discuss the proof of this conjecture, which extends Croot's method, and uses Fourier analysis coupled with elementary number theoretic and combinatorial arguments. We will also review several still open conjectures concerning unit fractions. Salle 15-25-502 Adresse Jussieu