Analysis on a Fractal Set

TL;DR

This paper introduces a new non-archimedean analysis on Cantor sets using ultrametric valuation, defining a scale-invariant measure and derivative, with potential applications in number theory and mathematics.

Contribution

It develops a novel scale-invariant real analysis framework on Cantor sets using non-archimedean valuations and infinitesimals, extending classical analysis.

Findings

Valued measure equals the Hausdorff measure of the set.

Real functions gain new asymptotic properties under the scale-invariant derivative.

A scale-invariant analysis framework replaces classical derivatives with logarithmic derivatives.

Abstract

The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant ultrametric valuation of the form $\log_{\varepsilon^{-1}} (\varepsilon/x) $ for a given scale $\varepsilon>0$ and infinitesimals $0<x<\varepsilon, x\in I\backslash C$. Using this new absolute value, a valued (metric) measure is defined on $C $ and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton $\{0\}$ of the real line $R$ is replaced by a zero measure Cantor set. The Cantor function is realised as a locally constant function in this setting. The ordinary derivative $dx/dt$ in $R$ is replaced by the scale invariant logarithmic derivative $d\log x/d\log t$ on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.