# On H\"{o}lder solutions to the spiral winding problem

## TL;DR

This paper investigates the regularity of functions mapping line segments onto spirals with polynomial winding, establishing sharp H"older exponent results and analyzing the dimension theory of these spirals, including Assouad spectrum insights.

## Contribution

It provides the first sharp H"older regularity results for spiral mappings with polynomial winding and compares these with dimension-based bounds, highlighting the role of Assouad spectrum.

## Key findings

- Sharp H"older exponents for spiral maps established
- Assouad spectrum offers the best dimension-based bounds
- Assouad spectrum distinguishes different polynomial winding rates

## Abstract

The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret `regularity' in terms of H\"{o}lder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp H\"{o}lder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the H\"{o}lder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under H\"{o}lder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only `dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime.