On polynomial progressions via transference

We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over Z/NZ with N prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial P(y)∈ Z[y] with $P(0) = 0$, that the largest subset A⊆ [N] avoiding the pattern \[x, x+P(y),…, x+ kP(y)\] has size bounded by P,kN(logloglog N)^-ΩP,k(1).