We study the question on how large the size of singular component of translates of product probability measures can be in terms of Fisher information. We then prove some shift inequalities.
S. G. Bobkov. "The Size of Singular Component and Shift Inequalities." Ann. Probab. 27 (1) 416 - 431, January 1999. https://doi.org/10.1214/aop/1022677267