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

Julia Ruscher

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arXiv · 1203.0753

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

Julia Ruscher

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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.

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Julia Ruscher. 2012-03-04. Cantor set zeros of one-dimensional Brownian motion minus Cantor function. https://arxiv.org/abs/1203.0753

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