On Zaremba's Conjecture

Let the reduced rational \(p/q\) in \([0,1)\) have continued fraction expansion \([0;c_1,c_2,...,c_m]\). Let \(f(p,q)\) be the maximum of the \(c_i\), and let \(F(q)\) be the minimum value of \(f(p,q)\) as \(p\) ranges. Zaremba conjectured in 1972 that \(F(q) \leq 5\) for all \(q\), and that all but finitely many \(q\) have \(F(q)\leq2\). We will present recent joint work with Jean Bourgain on this conjecture.

Alex Kontorovich