  In this paper, we prove some new thickness theorems with partial derivatives.
We give some applications. First, we give a simple criterion that can judge
whether two scaled Cantor sets have non-empty intersection. Second, we prove
under some checkable conditions that the continuous image of arbitrary
self-similar sets with positive similarity ratios is a closed interval, a
finite union of closed intervals or containing interior. Third, we prove an
analogous Erd\H{o}s-Straus conjecture on the middle-third Cantor set. Finally,
we consider the solutions to the Diophantine equations on fractal sets. More
specifically, for various Diophantine equations, we cannot find a solution on
certain self-similar sets, whilst for the Fermat's equation, which is
associated with the famous Fermat's last theorem, we can find infinitely many
solutions on many self-similar sets.
