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Transactions of the American Mathematical Society

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Restrictions of Hölder continuous functions


Authors: Omer Angel, Richárd Balka, András Máthé and Yuval Peres
Journal: Trans. Amer. Math. Soc. 370 (2018), 4223-4247
MSC (2010): Primary 26A16, 26A45, 28A78, 54E52, 60G17, 60G22, 60J65
DOI: https://doi.org/10.1090/tran/7126
Published electronically: February 8, 2018
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Abstract: For $ 0<\alpha <1$ let $ V(\alpha )$ denote the supremum of the numbers $ v$ such that every $ \alpha $-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $ v$. Kahane and Katznelson (2009) proved the estimate $ 1/2 \leq V(\alpha )\leq 1/(2-\alpha )$ and asked whether the upper bound is sharp. We show that in fact $ V(\alpha )=\max \{1/2,\alpha \}$. Let $ \dim _{\mathcal {H}}$ and $ \overline {\dim }_{\mathcal {M}}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $ V(\alpha )$ is a consequence of the following theorem. Let $ \{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $ \alpha $. Then, almost surely, there exists no set $ A\subset [0,1]$ such that $ \overline {\dim }_{\mathcal {M}} A>\max \{1-\alpha ,\alpha \}$ and $ B\colon A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $ A\subset [0,1]$ such that $ \overline {\dim }_{\mathcal {M}} A>1-\alpha $ and $ B\colon A\to \mathbb{R}$ is $ \beta $-Hölder continuous for some $ \beta >\alpha $. The zero set and the set of record times of $ B$ witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic $ \alpha $-Hölder continuous functions.

Finally, let $ \{\mathbf {B}(t): t\in [0,1]\}$ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set $ D\subset [0,1]$ such that $ \dim _{\mathcal {H}} D\geq 1/3$ and $ \mathbf {B}\colon D\to \mathbb{R}^2$ is non-decreasing in each coordinate. It remains open whether $ 1/3$ is best possible.


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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: angel@math.ubc.ca

Richárd Balka
Affiliation: Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
Email: balka@math.ubc.ca

András Máthé
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: a.mathe@warwick.ac.uk

Yuval Peres
Affiliation: Microsoft Research, 1 Microsoft Way, Redmond, Washington 98052
Email: peres@microsoft.com

DOI: https://doi.org/10.1090/tran/7126
Keywords: Fractional Brownian motion, H\"older continuous, restriction, bounded variation, Hausdorff dimension, box dimension, Minkowski dimension, self-affine function, generic, typical, Baire category
Received by editor(s): April 19, 2015
Received by editor(s) in revised form: November 11, 2016
Published electronically: February 8, 2018
Additional Notes: The first author was supported in part by NSERC
The second and third authors were supported by the National Research, Development and Innovation Office-NKFIH, 104178
The third author was also supported by the Leverhulme Trust.
Article copyright: © Copyright 2018 American Mathematical Society

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