Self-Similar Fractals in Arithmetic

Details Self-Similar Fractals in Arithmetic Self-Similar Fractals in Arithmetic

2004-04-27

arxiv.org/abs/math/0404498v2

Mathematics

Number Theory

15 pages

Scientific paper

We define a notion of self-similarity on algebraic varieties by considering algebraic endomorphisms as "similarity" maps. Self-similar objects are called fractals, for which we present several examples and define a notion of dimension in many different contexts. We also present a strong version of Roth's theorem for algebraic points on a variety approximated by fractal elements. Fractals provide a framework in which one can unite several important conjectures in Diophantine geometry.

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