Fourier coefficient lower-bound conjecture for rank one transformations

Let $T$ be a rank one transformation with a fixed sequence of tower partitions $(\xi_n)$ such that $\xi_{n+1}$ refines $\xi_n$ and every measurable set $A$ can be approximated by a sequence of $\xi_n$-measurable sets $A_n$. For a nonzero, zero-mean, $\xi_{n_0}$-measurable function $f$, let $\sigma_f$ be its spectral measure and let $c_n$ denote its Fourier coefficients. Fourier coefficient lower-bound conjecture. The coefficients satisfy

$$\limsup_{n \to \infty} \frac{\log|c_n|}{n} \ge -\frac12.$$

This asserts a universal restriction on the exponential decay of Fourier coefficients of spectral measures associated with finite-stage measurable functions in rank one systems. The surrounding discussion notes that the precise possible decay rate for rank one spectral measures is not known, while power-decay bounds do not by themselves determine whether the spectral measure has a Lebesgue component.