  The concept of self-similarity on subsets of algebraic varieties is defined
by considering algebraic endomorphisms of the variety as `similarity' maps.
Self-similar fractals are subsets of algebraic varieties which can be written
as a finite and disjoint union of `similar' copies. Fractals provide a
framework in which, one can unite some results and conjectures in Diophantine
geometry. We define a well-behaved notion of dimension for self-similar
fractals. We also prove a fractal version of Roth's theorem for algebraic
points on a variety approximated by elements of a fractal subset. As a
consequence, we get a fractal version of Siegel's theorem on finiteness of
integral points on hyperbolic curves and a fractal version of Falting's theorem
on Diophantine approximation on abelian varieties.
