A uniqueness theorem for Riesz potentials

The existence is proved of a nonzero Hölder function $f\colon\mathbb{R}\to\mathbb{R}$ that vanishes together with its M. Riesz potential $f\ast\frac{1}{|x|^{1-\alpha}}$ at all points of some set of positive length. This result improves that of D. Beliaev and V. Havin.