Cantor set zeros of one-dimensional Brownian motion minus Cantor function

TL;DR

This paper investigates the zeros of one-dimensional Brownian motion minus generalized Cantor functions, revealing conditions under which zeros almost surely do not occur within certain Cantor sets, extending previous results on middle-α Cantor sets.

Contribution

It introduces a generalized class of Cantor functions with variable interval sizes, showing that many such functions lead to no zeros in the associated Cantor sets when added to Brownian motion.

Findings

Zeros in the Cantor set occur with positive probability only for specific Cantor functions.

Most generalized Cantor functions do not produce zeros in their Cantor sets when added to Brownian motion.

The results extend understanding of the interplay between Brownian motion and fractal sets.

Abstract

It was shown by Antunovi\'{c}, Burdzy, Peres, and Ruscher that a Cantor function added to one-dimensional Brownian motion has zeros in the middle $\alpha$-Cantor set, $\alpha \in (0,1)$, with positive probability if and only if $\alpha \neq 1/2$. We give a refined picture by considering a generalized version of middle 1/2-Cantor sets. By allowing the middle 1/2 intervals to vary in size around the value 1/2 at each iteration step we will see that there is a big class of generalized Cantor functions such that if these are added to one-dimensional Brownian motion, there are no zeros lying in the corresponding Cantor set almost surely.