The Zaremba conjecture in number theory is that every natural
number occurs as
the denominator of a rational number arising from a finite continued
fraction with digits
taking only values from $\{1,2,3,4,5\}$. This conjecture remains open, but
it was shown by Bourgain-Kontorovich
and Huang to be true for most natural numbers (in a density one sense).
Interestingly, the proof relies
on a specific Cantor set X in the real line having (Hausdorff) dimension
greater than 5/6.
We will describe how to get this rigorous bound using the simple
dynamics of the Gauss map
defined by $T(x) = 1/x \pmod 1$. Time permitting, we will discuss other
related applications.