On binary forms II

The following joint paper of myself and my advisor Professor C.L. Stewart has been released on the arxiv. In this follow-up post, I would like to describe in some detail how to establish an asymptotic formula for the number of integers in an interval which are representable by a fixed binary form $F$ with integer coefficients and non-zero discriminant.

There are essentially three ingredients which go into the proof, each established decades apart. The first essential piece of the puzzle was established by Kurt Mahler in the 1930’s. He showed that if we examine the number of integer points in the region $\{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq Z\}$, then the number of such points is closely approximated by the area of the region. Since the region is homogeneously expanding, the area itself is well-approximated by scaling the fundamental region’ given by $\{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}$. Indeed, let $A_F$ denote the area of this fundamental region and let $N_F(Z)$ denote the number of integer pairs $(x,y) \in \mathbb{Z}^2$ such that $|F(x,y)| \leq Z$. Then Mahler proved that $\displaystyle N_F(Z) \sim A_F Z^{\frac{2}{d}}.$

More precisely, he proved a very good error term. He showed that when $d \geq 3$, we have $\displaystyle N_F(Z) = A_F Z^{\frac{2}{d}} + O_F\left(Z^{\frac{1}{d-1}} \right).$

The question then becomes is there some way to remove the redundancies in Mahler’s theorem? For example, if $F$ has even degree, then $F(x,y) = F(-x,-y)$ for all $(x,y) \in \mathbb{R}^2$, so the pairs $(x,y), (-x,-y) \in \mathbb{Z}^2$ represent the same integer. Is it true that this is the only way that this can happen? Unfortunately, the answer is no. For example, consider the binary form $F_n = x^n + (x - y)(2x - y) \cdots (nx-y)$. Then clearly the points $(1,1), \cdots, (1,n)$ all represent 1, and this construction works for any positive integer $n$. Therefore, there does not appear to be a simple way to count the multiplicities of points representing the same integer in Mahler’s theorem.

While examples like the above exist, perhaps it is possible that this happen sufficiently rare as to be negligible. For instance, if only $O\left(Z^{2/d - \delta}\right)$ many points counted by $N_F(Z)$ are such that there exist many essentially different’ (precise definition to come) other points which represent the same integer, and even in the worst case there can be at most say $O\left(Z^{\frac{\delta}{2}}\right)$ many essentially different pairs, then we have shown that in total, the contribution from these bad points to $N_F(Z)$ is only $O\left(Z^{\frac{2}{d} - \frac{\delta}{2}}\right)$, which is fine.

We shall now make some definitions. We say that an integer $h$ is essentially represented by $F$ if there exist two integer pairs $(x_1, y_1), (x_2, y_2)$ for which $F(x_1, y_1) = F(x_2, y_2) = h$, then there exists an element $\displaystyle T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \text{GL}_2(\mathbb{Q})$ such that $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = T \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$

and such that $F_T(x,y) = F(t_1 x + t_2 y, t_3 x + t_4 y) = F(x,y)$

for all $(x,y) \in \mathbb{C}^2$. Otherwise, we say that $h$ is not essentially represented.

Now put $R_F(Z)$ to be the number of integers up to $Z$ which are representable by $F$, and let $R_F^{(1)}(Z)$ be the number of essentially represented integers and $R_F^{(2)}(Z)$ be the number of non-essentially represented integers. If we can show that $R_F(Z) \sim R_F^{(1)}(Z)$, then we are basically done. This amounts to showing that $R_F^{(2)}(Z)$ is small compared to $Z^{\frac{2}{d}}$.

Christopher Hooley proved this for both the ‘easy cubic case’ and the ‘hard cubic case’. However, it was D.R. Heath-Brown who showed that $R_F^{(2)}(Z)$ is always small compared to $R_F^{(1)}(Z)$. This paved the way to our eventual success at this problem.

It remains to account for the interaction between those $T \in \text{GL}_2(\mathbb{Q})$  which fix $F$ and $R_F^{(1)}(Z)$. These elements are called the rational automorphisms of $F$ and we denote them by $\text{Aut} F = \text{Aut}_\mathbb{Q} F$. The most novel contribution we made to this topic is that we accounted for the exact interaction between $\text{Aut} F$ and $R_F^{(1)}(Z)$ with the so-called ‘redundancy lemmas’. This will be discussed at a future time.