Large gaps between values of several binary quadratic forms

Abstract

In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_1$, $D_2,\dots ,D_r$ be negative integers and $(s_n{)}_{n=1}^{\infty }$ be the sequence of all the numbers representable by any binary quadratic form of discriminant $D_1$, $D_2$, $\dots$ or $D_r$, and let $d:=\mathrm{lcm}\{D_1,\dots ,D_r\}$. We show that then \begin{align*} \limsup_{n\to\infty}\frac{s_{n+1}-s_{n}}{\log s_{n}}\geq \frac{1}{\log d + \log\log d + \log\log\log d + 4}. \end{align*} This improves and generalises a result by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard. As a by-product of our preliminary results, we show an improvement to the Pólya-Vinogradov inequality.