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.