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4. Fourier uniformity


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      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*}
        1. Remark. [J. Teräväinen] In [https://arxiv.org/abs/2007.15644], Matomäki, Radziwiłł, Tao, Teräväinen and Ziegler proved this for H\geq \exp((\log X)^{5/8+\varepsilon}) for any \varepsilon>0.


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              Problem 4.2.

              Can we extend the local Fourier uniformity conjecture to nilsequences? Warm-up: do it for polynomial phases.
                1. Remark. [J. Teräväinen] In [https://arxiv.org/abs/2007.15644], Matomäki, Radziwiłł, Tao, Teräväinen and Ziegler obtained this extension in the regime H\geq X^{\varepsilon} for any \varepsilon>0.


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                      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.