The Waring-Goldbach Problems with Almost Equal Variables

The investigation of Waring-Goldbach problem goes back to L.-K. Hua in 1938. This problem asks whether sufficiently large number

$n$ satisfying local conditions can be expressed as a sum of powers of primes. An intensively studied refinement of Hua’s theorem is that in which the variables are constrained to be almost equal, i.e., lied in a short interval $I_\theta =[x-x^\theta,x + x^\theta]$, where $x=(n/s)^{1/k}$ and $\theta \in (1/2,1)$ In this talk, I will present my recent work on this problem. The proof makes heavy use of “transference principle”. We also show results with exceptional sets by establishing almost all version of the transference principle.