Quantitative Gowers uniformity of the primes in intervals of length $X^{5/8+\varepsilon}$

Abstract

We prove the first quantitative bounds for the Gowers norms of the von Mangoldt function minus a Siegel corrected model function in short intervals. The result applies to intervals $(X,X+H]$ with $H\geq X^{5/8+\varepsilon}$ and gives quasipolynomial savings for the $U^k(X,X+H]$ norm. The main new ingredient is an efficient non-abelian Type II inverse theorem with explicit dependence on the nilsequence dimension. We prove this inverse theorem by applying Leng's efficient equidistribution theorem directly to an unweighted corner correlation in four variables, hence replacing the factorisation and subgroup argument in the non-abelian Type II proof of Matom\"aki, Shao, Tao and the author. In joint work with Florian Richter, we use this quantitative result as one ingredient in proving the existence of infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ in the shifted primes $\mathbb{P}-1$.