Year: 2026
Type: preprint
Abstract: We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequencies over short intervals. As an application, we prove that \[ \Big|\sum_{x0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|qα\| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials on the critical range would lead to a corresponding improvement in the lower bound for $H$.
Source: arXiv (Cornell University)
Authors: Pliego, Javier, Wang, Mengdi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221176578