84 citations · 187 across the 10 of their papers we have counts for
14 papers
The Siren Call of Silicon Leviathan: Reflections on blowup and Aufklärungsdämmerung
Alexander Gamburd
Over three centuries ago our Enlightenment forefathers brought forth in Europe--and later on this continent--a new approach to knowledge production and dissemination, conceived in…
On indefinite integral ternary quadratic forms
Alexander Gamburd, Amit Ghosh, Peter Sarnak +1
We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop…
Markoff Surfaces and Strong Approximation: 1
Jean Bourgain, Alexander Gamburd, Peter Sarnak
We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli.
An asymptotic formula for integer points on Markoff-Hurwitz varieties
Alex Gamburd, Michael Magee, Ryan Ronan
We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k. \] When $n…
Markoff Triples and Strong Approximation
Jean Bourgain, Alex Gamburd, Peter Sarnak
We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Mark…
A Spectral Gap Theorem in
Jean Bourgain, Alex Gamburd
We establish the spectral gap property for dense subgroups of (), generated by finitely many elements with algebraic entries; this result was announced in [BG3]. T…