Georgia Institute of Technology College of Sciences

The classical Wiener-Wintner Theorem says that for all measure preserving systems, and bounded functions f, there is a set of full measure so that the averages below converge for all continuous functions  g from the circle (R/Z)  to the complex numbers.

N^{-1} \sum_{n=1}^N  g( \pi n) f(T^n). 

We extend this result to averages over the prime integers. The proof uses structure of measure preserving systems, higher order Fourier analysis, and the Heath-Brown approximate to the von Mangoldt function.  A key result is a surprisingly small  Gowers norm estimate for the Heath-Brown approximate with fixed height.  

Joint work with  Y. Chen, A. Fragkos,  J. Fornal, B. Krause, and H. Mousavi.