I wish to establish the following estimate:
Conjecture :(correlation of Mobius function and nil-sequences in short interval)
$ \lambda(n)$ is the liouville function we wish the following estimate is true.
$ \int_{0\leq x\leq X}|\sup_{f\in \Omega^m}\sum_{x\leq n\leq x+H}\lambda(n)e^{2\pi if(x)}|dx =o(XH)$.
Where we have $ H\to \infty$ as $ x\to \infty$, $ \Omega^m=\{a_mx^m+a_{m-1}x^{m-1}+…+a_1x+a_0 | a_m,…,a_1,a_0\in [0,1]\}$ is a compact space.
I do not know how to prove this but this is result is valuable to consider, because by a Fourier identity we could transform the difficulty of (log average) Chowla conjecture to this type of result.
There is some clue to show this type of result could be true, the first one is the result established by Matomaki and Raziwill in 2015:
Theorem (multiplication function in short interval)
$ f(n): \mathbb N\to \mathbb C$ is a multiplicative function, i.e. $ f(mn)=f(n)f(m), \forall m,n\in \mathbb N$. $ H\to \infty$ as $ x\to infty$, then we have the following result,
$ \int_{1\leq x\leq X}|\sum_{x\leq n\leq x+H}f(n)|=o(XH)$.
And there also exists the result which could be established by Vinagrodov estimate and B-S-Z critation :
Theorem(correlation of multiplication function and nil-sequences in long interval)
$ f(n): \mathbb N\to \mathbb C$ is a multiplicative function, i.e. $ f(mn)=f(n)f(m), \forall m,n\in \mathbb N$. $ g(n)=a_n^m+…+a_1n+a_0$ is a polynomial function then we have the following result,
$ \int_{1\leq n \leq X}|f(n)e^{2\pi i g(n)}|=o(X)$.