My Research

In 2013 Thomas Jordan and Tuomas Sahlsten [JS] proved that if a measure is statistical (Gibbs) with respect to the Gauss map, and that the dimension of the measure is bigger than a half, then the Fourier transform of the measure has polynomial decay. This paper had great consequence because it was used to prove a long-standing problem of Salem posed in 1943. The theorem also attracted the interest of the Dynamics community by showing that we could soon improve our understanding of how Fourier decay relates to nonlinear invariance.

Before we could start studying this relation, Tuomas and I first generalised Jordan-Sahlsten by reducing the dimension assumption. Both authors believed that this assumption was unnecessary and was introduced only for technical reasons. By using the techniques presented by Jean Bourgain and Semyon Dyatlov in 2017 [BD], we could reduce this assumption to requiring that the dimension of the measure is nonzero. In the zero case, it is likely that different techniques will be required.

By removing the dimension assumption, we see that the nonlinear map invariance is the driving force behind the polynomial Fourier decay. This is because we reduce to proving a statement which says that the derivatives of the branches of the map distributed well. This was proved in the Gauss map case using the methods of Queffélec and Ramaré [QR] on Diophantine approximation which were used in Jordan-Sahlsten.

The question is whether this could be generalised to a large class of nonlinear maps. In 2020 Tuomas and I published follow up work which shows that we can generalise to totally nonlinear maps. When considered in conjunction of the works of Carolina Mosquera and Pablo Shmerkin [MS] in 2017, which considers the not totally nonlinear case, we get a somewhat complete picture of this problem when considering one dimensional maps.