Joint IAS/Princeton University Number Theory Seminar

In a recent work, Matomäki, Radziwill and Tao showed that the Möbius function is discorrelated with linear exponential phases on almost all intervals of length $X^{\varepsilon}$. I will discuss joint work where we generalize this result to nilsequences, so as a special case the Möbius function is shown not to correlate with polynomial phases on almost all intervals of length $X^{\varepsilon}$. As an application, we show that the number of sign patterns of length $k$ that the Liouville function takes grows superpolynomially in $k$.