# Fourier decay for self-similar measures

## TL;DR

This paper proves that most self-similar measures on the line exhibit power decay in their Fourier transform at infinity, extending classical results to non-homogeneous cases with complex contraction ratios.

## Contribution

It establishes Fourier decay for a broad class of self-similar measures, including non-homogeneous cases, outside a zero Hausdorff dimension set of parameters.

## Key findings

- Most self-similar measures have Fourier transform decay at infinity.
- Fourier decay holds outside a zero Hausdorff dimension exceptional set.
- Extends classical results from homogeneous to non-homogeneous measures.

## Abstract

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erd\H{o}s and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.