Longer gaps between values of binary quadratic forms

TL;DR

This paper investigates the size of gaps between numbers represented by binary quadratic forms, establishing new lower bounds that grow with the discriminant and improving previous results, especially for sums of two squares.

Contribution

It provides improved lower bounds on gaps between representable numbers, including a specific enhancement for sums of two squares, and extends results to sparse sequences.

Findings

Lower bounds on gaps grow with the discriminant |D|

Improved bound for sums of two squares to approximately 0.434

Generalized results to sparse sequences with large gaps

Abstract

Let $s_1, s_2, \ldots$ 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 \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \ge \frac{\varphi(|D|)}{2|D|(1+\log \varphi(|D|))}\gg \frac{1}{\log \log |D|}, \] 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 $\frac{195}{449}=0.434\ldots$. 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.