  We study when Fourier transforms of Gibbs measures of sufficiently nonlinear
expanding Markov maps decay at infinity at a polynomial rate. Assuming finite
Lyapunov exponent, we reduce this to a nonlinearity assumption, which we verify
for the Gauss map using Diophantine analysis. Our approach uses large
deviations and additive combinatorics, which combines the earlier works on the
Gibbs measures for Gauss map (Jordan-Sahlsten, 2013) and Fractal Uncertainty
Principle (Bourgain-Dyatlov, 2017).
