Fourier decay for curved Frostman measures

Abstract

We investigate Fourier decay for Frostman measures supported on curves with nonzero curvature. We combine decoupling with known lower bounds for Furstenberg sets to extend the main result of Orponen [Ann. Fenn. Math. 48 (2023), pp. 113–139].

Pointwise normality and Fourier decay for self-conformal measures

“…The family of homomorphic functions log h N k ,ω is pre-compact by (19), whence there exists a holomorphic function…”

“…However, in the presence of some curvature the situation becomes much more interesting. Here, following the initial work of Orponen [46], there have been several papers [19,21,47] that prove quantitative Fourier decay estimates on average for (not necessarily dynamically defined) measures supported on such curves; however, we are not aware of such examples where pointwise estimates have been established.…”