In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak is a statement on the spectral gaps of certain actions of a free group on the sphere .
Any matrix defines an isometry of the sphere , which in turn defines an operator on the Hilbert space . The spectral gap conjecture states that for any integer , if isometries are chosen uniformly at random, then the operator has a nontrivial spectral gap with probability 1.
In 2007, Jean Bourgain and Alex Gamburd proved that when the matrices have entries which are all algebraic numbers up to simultaneous conjugation, the resulting operator has a spectral gap. This result was later generalized to the case of . It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1. If true, the statement would have applications to quantum computing and the design of universal quantum gate sets.