In this talk, we will begin with simple questions of the Diophantine Approximation Theory, for instance, how closely can a given irrational number x be approximated by a rational number r with denominator no larger than a fixed number? This will lead us to talk about the set known as the Lagrange Spectrum whose structure closely resembles the structure of the sum of dynamically defined Cantor sets, which are defined by an iterate system of expanding differentiable functions on intervals. We will construct two Cantor sets whose arithmetic sum is a uniformly contracting self-similar set. A local result on a sufficient condition for a uniformly contracting self-similar set to be of Lebesgue measure zero will be proven.