Large gaps between values of several binary quadratic forms

In this paper we study the problem of long gaps between values of binary quadratic forms. Let D₁, D₂,… ,Dᵣ be negative integers and (sₙ)ₙ₌₁∞ be the sequence of all the numbers representable by any binary quadratic form of discriminant D₁, D₂, … or Dᵣ, and let d := lcm₁,… ,Dᵣ\. We show that then {align*} n→∞{sₙ₊₁-sₙ}{log sₙ}≥ 1/log d + loglog d + logloglog d + 4. {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.