Self-similar measures and the Rajchman property
Julien Brémont
Abstract
For general self-similar measures associated with contracting on average affine IFS on the real line, we study the convergence to zero of the Fourier transform at infinity (or Rajchman property) and the extension of results of Salem [Sal44] and Erdös [Erd39] on Bernoulli convolutions. Revisiting in a first step a recent work of Li–Sahlsten [LS19], we show that the parameters where the Rajchman property may not hold are very special and in close connection with Pisot numbers. In these particular cases, the Rajchman character appears to be equivalent to absolute continuity and, when the IFS consists of strict contractions, we show that it is generically not true. We finally provide rather surprising numerical simulations and an application to sets of multiplicity for trigonometric series.