
## 4. Fourier uniformity

1. #### Problem 4.1.

How much can we lower $H$ in the local Fourier uniformity conjecture \begin{equation*} \int_X^{2X} \sup_\alpha \left| \sum_{x\leq n\leq x+H} \lambda(n) e(n\alpha) \right| d x = o(HX)? \end{equation*}
• #### Problem 4.2.

Can we extend the local Fourier uniformity conjecture to nilsequences? Warm-up: do it for polynomial phases.
• #### Problem 4.3.

Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?

Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.