Longer gaps between values of binary quadratic forms

Abstract

Let $s_1,s_2,\dots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \ge \frac{φ(|D|)}{2|D|(1+\log φ(|D|))}\gg \frac{1}{\log \log |D|}, \] improving a lower bound of $\frac{1}{|D|}$ of Richards (1982). In the special case of sums of two squares, we improve Richards's bound of $1/4$ to $\frac{195}{449}=0.434\dots$. We also generalize Richards's result in another direction and establish a lower bound on long gaps between sums of two squares in certain sparse sequences.

14 pages; this version from 2018 will not be published in this form. It will appear in a joint and expanded manuscript by R. Dietmann, C. Elsholtz, A. Kalmynin, S. Konyagin and J. Maynard