Explicit upper bounds of Fourier transforms of non-Liouville self-similar measures

Abstract

We establish explicit decay rates for the Fourier transforms of self-similar probability measures supported on non-singleton self-similar sets, under a non-Liouville condition on a logarithmic ratio of contraction ratios. The decay exponent is given in closed form as a function of any chosen upper regularity exponent and the non-Liouville degree, turning the implicit bounds of Li–Sahlsten into fully effective estimates. In the presence of pairwise disjoint images, we obtain explicit decay for arbitrary self-similar measures, and identify a canonical one achieving the maximal decay permitted by our method. We then apply the theory to Lüroth digit-restricted sets, deriving computable exponents from explicit lower bounds for linear forms in logarithms and giving a worked numerical example for the two-digit case.