# On measures that improve $L^q$ dimension under convolution

## TL;DR

This paper investigates conditions under which the $L^q$ dimension of measures on the real line increases when convolved, showing that uniformly perfect measures cause a strict improvement, with implications for sumsets and fractal dimensions.

## Contribution

The paper establishes that uniformly perfect measures cause a strict increase in $L^q$ dimension under convolution, extending understanding of $L^p$-improving properties for a broad class of measures.

## Key findings

- Uniformly perfect measures increase $L^q$ dimension upon convolution.
- Results apply to repeated convolutions and sumset dimensions.
- Derived from an inverse theorem for $L^q$ norms of convolutions.

## Abstract

The $L^q$ dimensions, for $1<q<\infty$, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the $L^q$ dimension improve under convolution? This can be seen as a variant of the well-known $L^p$-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property that convolving with them results in a strict increase of the $L^q$ dimension. We also study the case $q=\infty$, which corresponds to the supremum of the Frostman exponents of the measure. We obtain consequences for repeated convolutions and for the box dimension of sumsets. Our results are derived from an inverse theorem for the $L^q$ norms of convolutions due to the second author.