# Oscillation of Functions in the H\"older class

## TL;DR

This paper investigates the size of sets where functions in the Hölder class exhibit significant oscillation, using dyadic martingale techniques to derive sharp Hausdorff measure estimates.

## Contribution

It introduces a novel approach linking Hölder function oscillations to dyadic martingale behavior, providing sharp measure bounds for oscillation sets.

## Key findings

- Established bounds on the Hausdorff measure of oscillation sets.
- Connected Hölder class properties with dyadic martingale growth.
- Provided sharp estimates for maximal growth sets in martingale models.

## Abstract

We study the size of the set of points where the $\alpha$-divided difference of a function in the H\"older class $\Lambda_\alpha$ is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.