Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along the primes, which is the natural pointwise convergence along prime result after Wierdl (1990) and a natural follow up after recent pointwise bilinear break-through by Krause Mirek-Tao (2021). In another project, the sharpest pointwise convergence average along primes for “near” integrable function is given, which is sharp up to Generalized Riemann Hypothesis.