Resumen de On measures that improve Lq dimension under convolution

Eino Rossi, Pablo Shmerkin

• The Lq dimensions, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the Lq dimension improve under convolution? This can be seen as a variant of the well-known Lp-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 Lq dimension. We also study the case q=∞, 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 Lq norms of convolutions due to the second author.