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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.