  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.
