Greatest common divisors and Diophantine approximation

In 2003, Bugeaud, Corvaja, and Zannier gave an (essentially sharp) upper bound for the greatest common divisor gcd(a^n-1,b^n-1), where a and b are fixed integers and n varies over the positive integers.  In contrast to the elementary statement of their result, the proof required deep results from Diophantine approximation.  I will discuss a higher-dimensional generalization of their result and some related problems, all centered around greatest common divisors.