The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially geometry Euclidean spaces. We survey these developments, emphasis on connections computable functions the reals, recent uses proving new theorems classical (non-algorithmic) geometry, and directions for future research.

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the tubular neighborhood of a p-adic fractal string Lp, expressed in terms of th...

The Invariant Set Hypothesis proposes that states of physical reality belong to, and are governed by, a non-computable fractal subset I of state space, invariant under the action of some subordinate deterministic causal dynamics I D . The Invariant Set Hypothesis is motivated by key results in nonlinear dynamicalsystems theory, and black-hole thermodynamics. The elements of a reformulation of q...