Quantitative Gowers uniformity of the primes in intervals of length X5/8+ε
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≥ X5/8+ε and gives quasipolynomial savings for the Uk(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 Matomäki, 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 ℙ−1.
Related
- Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals
- Bounded intervals containing a given number of primes
- Biases in the distribution of primes in short intervals
- Quasimodular Forms, Partitions, and Primes in Intervals
- The Poisson Tail Conjecture for primes in short intervals
BibTeX
@misc{teravainen2026quantitative,
title = {{Quantitative Gowers uniformity of the primes in intervals of length $X^5/8+\varepsilon$}},
author = {Joni Teräväinen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03707},
eprint = {2610.03707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03707},
}
RIS
TY - UNPB
TI - Quantitative Gowers uniformity of the primes in intervals of length X^5/8+ε
AU - Joni Teräväinen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.03707
UR - https://arxiv.org/abs/2610.03707
AB - 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≥ X^5/8+ε 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 Matomäki, 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 ℙ−1.
ER -