The HADES seminar on Tuesday, **November 23rd**, will be given by **Yuqiu Fu** at **5 pm** on Zoom.

**Speaker**: Yuqiu Fu (MIT)

**Abstract**: If the Fourier transform of a function $f:\mathbb R \rightarrow \mathbb C$ is supported in a neighborhood of an arithmetic progression, then $|f|$ is morally constant on translates of a neighborhood of a dual arithmetic progression.

We will discuss how this “locally constant property” allows us to prove sharp decoupling inequalities for functions on $\mathbb R$ with Fourier support near certain convex/concave sequence, where we cover segments of the sequence by neighborhoods of arithmetic progressions with increasing/decreasing common difference. Examples of such sequences include $\{\frac{n^2}{N^2}\}_{n=N+1}^{N+N^{1/2}}$ and $\{\log n\}_{n=N+1}^{N+N^{1/2}}.$

The sequence $\{\log n\}_{n=N+1}^{2N}$ is closely connected to Montgomery’s conjecture on Dirichlet polynomials but we see some difficulties in studying the decoupling for $\{\log n\}_{n=N+1}^{2N}.$ This is joint work with Larry Guth and Dominique Maldague.