1 Introduction↩︎

For any integer \(h\), let \[\mathcal{D}_h = \{ (a,b,c,d) \in \mathbb{Z}^4 : ad - bc = h\}.\] One can naturally interpret \(\mathcal{D}_h\) as the set of \(2\times 2\) integer matrices with determinant \(h\). There are various classical questions in analytic number theory that concern counting elements of \(\mathcal{D}_h\) in expanding regions. For instance, writing \(\mathcal{B}(N)\) to be the ball of radius \(N\) in \(\mathbb{R}^4\) for every \(N \in \mathbb{N}\), we can define \[T_{\ell^2}(h,N) = |\mathcal{D}_h \cap \mathcal{B}(N)| = 6 N^2 (\sum_{d|h} 1/d) + E_{\ell^2}(h,N) .\] A classical result of Selberg states that \[\label{selbergresult} E_{\ell^2}(1,N) \ll N^{4/3}.\tag{1}\] This bound has not been improved, and in fact it is conjectured that \[\label{conjl2} E_{\ell^2}(1,N) \ll_{\varepsilon} N^{1 + \varepsilon},\tag{2}\] see [1]. A more modern version of 1 is recorded by Iwaniec [1], which implies that \(E_{\ell^2}(h,N)\ll h^{1/3} N^{4/3} \sum_{d|h}1/d\) for all \(1 \leqslant h \leqslant N^2\).

One can also consider counting \((a,b,c,d) \in \mathcal{D}_h\) with \(a,b,c,d \in \mathbb{N}\) and \(bc \leqslant N^2\). Thus, we define \[T_{\rm div}(h, N) = \sum_{1 \leqslant n \leqslant N^2} d(n) d(n+h) = M_{\rm div}(h,N) + E_{\rm div}(h,N),\] where \(d(n) = \sum_{x,y \in \mathbb{N}} \mathbb{1}_{n = xy}\) counts the number of divisors of \(n\) and \(M_{\rm div}(h,N)\) denotes the main term described in [2]. The problem of finding good quantitative estimates for the above is known as the binary additive divisor problem, see [2]–[6] and the references therein. This is a very well-studied problem, in part, due to its close connections to the fourth moment of the Riemann zeta function, see [7]. The work of Motohashi [2], which relies on spectral methods, implies that for any \(h \leqslant N^{40/27}\) one has \[E_{\rm div}(h,N) \ll_{\varepsilon} N^{4/3 + \varepsilon},\] see also [2] for a modified asymptotic formula which holds for a wider range of \(h\). Motohashi [2] also proved that for fixed \(h\) and large \(N\), one has \[\label{dnd4} E_{\rm div}(h,N) \gg N.\tag{3}\] It is natural to speculate that this should be almost optimal. In fact, Conrey–Gonek [8] have conjectured that for all \(0 < h \ll N\), one should have \[\label{divconj} E_{\rm div}(h,N) \ll_{\varepsilon} N^{1 + \varepsilon}.\tag{4}\]

From the perspective of counting integer solutions to quadratic equations, a natural question concerns studying points in \(\mathcal{D}_h \cap [-N,N]^4\). This type of problem arises in various contexts including combinatorial geometry [9], counting commuting matrices [10], [11], diophantine approximation [12], [13], theory of random multiplicative functions [14], [15], and counting integer solutions to quadratic equations [16], [17]. Thus, we define \[T(h,N) = |\mathcal{D}_h \cap [-N,N]^4| = \sum_{n \in \mathbb{Z}} d'(n) d'(n+h), \;\;\text{where} \;\;d'(n) = \sum_{|x|,|y| \leqslant N} \mathbb{1}_{n= xy}\] is the restricted divisor function. We further write \[\label{defthn} T(h,N) = \frac{16}{\zeta(2)} N^2 \big( \sum_{d|h}1/d \big) + E_{\rm \ell^{\infty}}(h,N) .\tag{5}\] This can be construed as a circle-method heuristic; indeed, the first term here can be written as a product of a singular series and a singular integral when \(h \ll_{\varepsilon} N^{2 - \varepsilon}\), see [18].

The problem of estimating \(T(h,N)\) was first analysed by Afifurrahman [19], who used results on counting points on modular hyperbola, which themselves require deep estimates of Weil for Kloosterman sums, to prove that for every \(1 \leqslant h \leqslant 2N^2\) one has \[\label{res1} E_{\rm \ell^{\infty}}(h,N) \ll_{\varepsilon}N^{\varepsilon}( h + N^{5/3}).\tag{6}\] Subsequently, Ganguly–Guria [20] used Fourier analysis and spectral methods to show that in the shorter range \(1 \leqslant h \leqslant N^{1/3}\) one has \[\label{res2} E_{\rm \ell^{\infty}}(h,N) \ll_{\varepsilon} h^{\theta} N^{3/2 + \varepsilon},\tag{7}\] where \(\theta\) is any admissible exponent for the generalised Ramanujan conjecture. Deep work of Kim–Sarnak [21] implies that \(\theta \leqslant 7/64\).

Noting the aforementioned bounds for \(E_{\ell^2}(h,N)\) and \(E_{\rm div}(h,N)\), it is reasonable to speculate that the bounds in 6 and 7 can be improved significantly. We are able to do this in a short and completely elementary fashion.

This improves upon 6 and 7 , while also confirming an analogue of the conjectured bounds 2 and 4 in our setting. Indeed, Theorem 1 implies that for all \(0 < h \leqslant N\) one has \[\label{errortermepsilon} E_{\ell^{\infty}}(h,N) \ll_{\varepsilon} N^{1 + \varepsilon}.\tag{8}\] We note that our proof can be analysed more carefully to give a more precise characterisation of the \(N^{\varepsilon}\) factor in the error term, but we have not pursued this here.

Our methods also deliver an analogue of Motohashi’s lower bound 3 for \(E_{\ell^{\infty}}(h,N)\) for much larger shifts \(h\).

Thus, whenever \(h \geqslant N^{1+ \delta}\) for any fixed \(\delta>0\), one cannot expect 8 to hold, and the error term in Theorem 1 turns out to be roughly of the right order.

An interesting aspect of this lower bound is that it holds in the range \(h \gg N^2\), which in turn means that the error term in 5 matches the order of the main term for many choices of \(h \gg N^2\). This does not mean that the circle-method heuristic for 5 fails completely when \(h \gg N^2\), it just means that one expects a different main term because the size of the singular integral varies as \(h\) varies, see [18]. In fact, the latter setting is much harder than the regime \(h = o(N^2)\) studied in this paper and nothing seems to have been previously known for \(T(h,N)\) when \(N^2 \ll h \leqslant(2 - o(1))N^2\). Moreover, the only known result for \(T_{\ell^2}(h,N)\) in this setting seems to follow from a general class of results proven by Oh [22] using techniques from dynamics. While Theorem 2 implies that our methods in this paper cannot yield non-trivial estimates for \(T(h,N)\) when \(h \gg N^2\), we use completely different methods to analyse this setting in [18].

We remark that obtaining an asymptotic formula in the case when \(h=0\) is significantly simpler than the cases when \(h \neq 0\) since the former allows for many additional symmetries. Indeed, writing \(T(0,N)\) to be the number of solutions to \(ad =bc\) with \(a,b,c,d \in [-N,N]\), one can swiftly show that \[\label{singularmatrixcount} T(0,N) = \frac{16}{\zeta(2)}N^2 \log N + O(N^2).\tag{9}\] An asymptotic formula with an explicit second order term of order \(N^2\) can be deduced from work of Ayyad–Cochrane–Zheng, see [23].

As previously mentioned, our results have a natural interpretation as counting integer matrices with fixed determinant and bounded \(\ell^{\infty}\) norm. Similarly, \(T_{\ell^2}(1,N)\) counts matrices \(\gamma \in {\rm SL}_2(\mathbb{Z})\) such that \(\|{\gamma}\| = {\rm Tr}(\gamma^T \gamma)^{1/2} \leqslant N\). The latter type of results have been generalised to matrices \(\gamma \in {\rm SL}_n(\mathbb{Z})\) with \(n \geqslant 3\), see, for instance, the very nice work of Duke–Rudnick–Sarnak [24], and subsequent results by Gorodnik–Nevo–Yehoshua [25] and Blomer–Lutsko [26]. A higher dimensional analogue of 9 in the bounded \(\ell^2\) case was established by Katznelson [27] via techniques from the geometry of numbers. It would also be interesting to consider versions of these higher dimensional results in the bounded \(\ell^{\infty}\) case.

Outline↩︎

We use §2 to record various standard lemmas which we will employ in our proof of Theorems 1. We begin our proof of Theorem 1 in §3 by performing some preliminary manoeuvres to reduce our problem to analysing a weighted count of points in arithmetic progressions with varying lengths and moduli. We are able to make these arithmetic progressions slightly more uniform in size at the cost of introducing the error term appearing in Theorem 1. In §4, we use this uniformisation trick to further reduce our problem to estimating pairs of coprime integers in large intervals and counting coprime residue classes satisfying extra congruence conditions, both of these being standard results from elementary number theory. We conclude §4 by proving Theorem 2.

Notation↩︎

We employ Vinogradov notation, that is, we write \(Y \ll_{z} X\), or equivalently \(Y =O_z(X)\), to mean that \(|Y| \leqslant C_z X\), where \(C_z>0\) is some constant depending on the parameter \(z\). Unless stated otherwise, whenever \(\varepsilon\) appears in any bound, it will mean that the bound holds for every \(\varepsilon>0\), though the implicit constant may depend on \(\varepsilon\). We denote the greatest common divisor of two integers \(a\) and \(b\) by \((a,b)\).

Acknowledgements↩︎

We thank Sam Chow, Lasse Grimmelt, Jori Merikoski, V. Vinay Kumaraswamy, and Trevor Wooley for helpful comments. JC is supported by EPSRC through Joel Moreira’s Frontier Research Guarantee grant, ref. EP/Y014030/1. AM is supported by a Leverhulme Early Career Fellowship ECF-2025-148.

Finally, as we were finishing the first version of this paper, it came to our attention that Dhanda–Haynes–Prasala [28] have independently proved Theorem 1.

Rights↩︎

For the purpose of open access, the authors have applied a Creative Commons Attribution (CC-BY) licence to any Author Accepted Manuscript version arising from this submission.

2 Auxiliary lemmas↩︎

We utilise this section to record various standard results from elementary number theory that we will use throughout our paper.

We will also require the following standard estimate on the number of elements of an interval which are coprime to some fixed moduli, see [29].

3 Preliminary manoeuvres↩︎

We employ this section and the next to present our proof of Theorem 1. Throughout this section, let \(N\) be some natural number and let \(1 \leqslant h \leqslant 2N^2\). Upon excluding solutions to \(ax - by = h\) with \(abxy = 0\) and noting the various symmetries, we see that \[T(h,N) = 4 \sum_{\substack{d|h, \\ d \leqslant N}} \big( \sum_{\substack{1 \leqslant u,v \leqslant N/d \\ (u,v) = 1}} \;\sum_{x,y \in [-N,N]} \mathbb{1}_{ux - vy = h/d} \big) + O_{\varepsilon}(N^{1 + \varepsilon}).\] Excluding further the contribution of the solutions when \(u = v = 1\), which contribute at most \(O_{\varepsilon}(N^{1 + \varepsilon})\) due to the divisor bound, and then noting that in all other cases we may assume without loss of generality that \(u >v\) (as \(x,y \in [-N,N]\)), we obtain \[\label{eqn3461} T(h,N) = 8 \sum_{\substack{d|h, \\ d \leqslant N}} \big( \sum_{\substack{1 \leqslant v < u \leqslant N/d \\ (u,v) = 1}} \;\sum_{x,y \in [-N,N]} \mathbb{1}_{ux - vy = h/d} \big) + O_{\varepsilon}(N^{1+\varepsilon}).\tag{10}\] Hence, given \(n\in \mathbb{N}\) and coprime integers \(1\leqslant v < u\), we define \[r_{u,v}(n) = \sum_{x,y \in [-N,N]} \mathbb{1}_{ux - vy = n} \;\;\text{and} \;\; \tilde{r}_{u,v}(n) = \sum_{y\in [-N,N]} \mathbb{1}_{y \equiv -n v^{-1} \;(\mathop{\mathrm{mod}}{ u}) }.\] Note that \(y \in [-N,N]\) can lead to a valid solution of \(ux - vy = n\) with \(x \in [-N,N]\) if and only if \[y \equiv -n v^{-1} \;(\mathop{\mathrm{mod}}{ u}) \;\;\text{and} \;\;y \in [-(Nu+n)/v, (Nu-n)/v] \cap [-N,N].\] Furthermore, since \(u/v >1\) and \(n \geqslant 1\), we always have \(-(Nu+n)/v < - N\). This immediately implies that \[\begin{align} \label{eqn3462} 0 \leqslant\tilde{r}_{u,v}(n) - r_{u,v}(n) & \leqslant\sum_{ (Nu -n)/v < y \leqslant N} \mathbb{1}_{y \equiv - n v^{-1} \;(\mathop{\mathrm{mod}}{ u})} \nonumber \\ & = \bigg(\frac{n - N(u-v)}{uv} + O(1) \bigg) \mathbb{1}_{n \geqslant N(u-v)} . \end{align}\tag{11}\] In fact, if \(1 \leqslant n < N(u-v)\), then \(r_{u,v}(n) = \tilde{r}_{u,v}(n)\), and if \(n > N(u+v)\), then \(r_{u,v}(n) = 0\) while \(|\tilde{r}_{u,v}(n) -2N/u| \leqslant 5\). Moreover, when \(N(u-v) \leqslant n \leqslant N(u+v)\), equality holds in the second inequality in 11 .

Setting \[\tilde{T}(h,N) = 8 \sum_{\substack{d|h, \\ d \leqslant N}} \big( \sum_{\substack{1 \leqslant v < u \leqslant N/d \\ (u,v) = 1}} \;\tilde{r}_{u,v}(h/d) \big) ,\] our proof of Theorem 1 now reduces to proving that \(\tilde{T}(h,N)\) is a good approximation for \(T(h,N)\) and then appropriately estimating \(\tilde{T}(h,N)\). The following lemma proves the former statement.

Thus, it suffices to prove Theorem 1 for \(\tilde{T}(h,N)\), and we will do so in the next section.

4 Proof of Theorems 1 and 2↩︎

Our first aim of this section is to prove the following result.

We remark that Lemma 6 combines with Lemma 5 to dispense the conclusion of Theorem 1. We now present the proof of Theorem 2.

References↩︎