1. Introduction

It is a well-studied problem to estimate the number $S_{k,\|\cdot\|}(n,X)$ of $k\times k$ matrices A with integer coefficients, $\det(A)=n$ , and $\|A\|\le X$ , where $k\ge 2$ , $n\in\mathbb{Z}$ , $X\ge 1$ , and $\|\cdot\|$ is a prescribed norm. For the Euclidean norm $\|\cdot\|_2$ , a famous result of Selberg from the 1970’s on the hyperbolic circle problem (see Selberg’s unpublished notes [ Reference Selberg17 , p. 5 of document 3]) gives that

\begin{equation*} S_{2,\|\cdot\|_2}(1,X)=6X^2+O_\epsilon\big(X^{4/3+\epsilon}\big),\end{equation*}

for any $\epsilon>0$ .

In 1993 Duke, Rudnick and Sarnak published a landmark paper on the density of integer points on affine homogeneous varieties [ Reference Duke, Rudnick and Sarnak6 ]. As one application of their results they proved (theorem 1.10 in their paper) that, for any $k\ge 2$ and for any $n\not= 0$ ,

(1·1) \begin{equation} S_{k,\|\cdot\|_2}(n,X)=c_{k,n}X^{k^2-k}+O_{n,\epsilon}\big(X^{k^2-k-1/(k+1)+\epsilon}\big),\end{equation}

for any $\epsilon>0$ , where $c_{k,n}$ is a constant determined by values of the gamma function, the Riemann zeta function, and the prime power divisors of n.

Analogous results for the case when $n=0$ were established by Katznelson in [ Reference Katznelson11 ] (cf. [ Reference Katznelson12 ]). These types of problems were further investigated, using tools from ergodic theory, by Eskin, Mozes and Shah [ Reference Eskin, Mozes and Shah7, Reference Shah18 ], who proved asymptotic formulas for numbers of matrices with bounded Euclidean norm and specified characteristic polynomial.

Surprisingly, the asymptotic behavior of $S_{k,\|\cdot\|_\infty}(n,X)$ , where $\|\cdot\|_\infty$ is the sup-norm, has been less well studied, even in the case when $k=2$ , until recently. The above mentioned work of Selberg, Duke, Rudnick and Sarnak relies on the rotational invariance of the Euclidean norm, and therefore does not apply to this problem. Asymptotic formulae for $S_{2,\|\cdot\|_\infty}(0,X)$ (with second order asymptotics) can be derived from work of Ayyad, Cochrane and Zhang [ Reference Ayyad, Cochrane and Zheng2 ]. There is an asymptotic formula for $S_{2,\|\cdot\|_\infty}(1,X)$ due to Roettger [ Reference Roettger16 ] (who also considers an analogous problem over rings of integers of algebraic number fields), as well as more recent work of Bulinski and Shparlinski [ Reference Bulinski and Shparlinski3 ], which leads to an improved error term (see [ Reference Afifurrahman1 , equation (1·4)]). A ‘smoothed’ version of the sup-norm problem for $k=2$ and arbitrary n (actually for $2\times 2$ matrices with a fixed characteristic polynomial) is considered in [ Reference Guria9 ], however adding smooth weights in the count does change the problem.

Significant progress on this problem was made by M. Afifurrahman, who in July 2024 published a preprint containing a proof of the following result.

Theorem [ Reference Afifurrahman1 , theorem 1·1]. For $n\in\mathbb{N}$ ,

(1·2) \begin{equation} S_{2,\|\cdot\|_\infty}(n,X)=\frac{96\sigma_{-1}(n)}{\pi^2}X^2+O\big(X^{o(1)}\max\big\{X^{5/3},n\big\}\big),\end{equation}

as $X,n\rightarrow\infty$ .

In the statement of the theorem, $\sigma_s(n)$ denotes the sum of the $s{\text{th}}$ powers of the positive divisors of n. One important feature of this result is that, in contrast to (1·1), the dependence on n of the error term here is made explicit, so that both X and n may tend to infinity as functions of one another. This is necessary for some applications (cf. [ Reference Shparlinski19 , theorem 4], [ Reference Ostafe and Shparlinski15 , section 2·2]), and the theorem is significant for X even as small as $n^{1/2+\delta}$ . For the case of large X, a preprint of Ganguly and Guria [ Reference Ganguly and Guria8 ] improves the error term in this theorem to $\ll_\epsilon n^\theta X^{3/2+\epsilon}$ (where $\theta$ is a non-negative constant), in the range where $X\gg n^3$ .

Our first aim in this paper is to give a short proof of the following theorem.

Theorem 1. Let $\epsilon>0$ . Then, for $n\in\mathbb{N}$ and $X\ge 1$ we have that

\begin{equation*} S_{2,\|\cdot\|_\infty}(n,X)=\frac{96\sigma_{-1}(n)}{\pi^2}X^2+O_\epsilon\left(\sigma_0(n)X\log X+nX^\epsilon\right). \end{equation*}

The bound in our theorem recovers the error term from (1·2) when X is small compared to n, but it gives a significant power savings (over both that result and the one in [ Reference Ganguly and Guria8 ]) when $X\gg n$ . To understand the dependence of the error term on n, note that

\begin{align*}\limsup_{n\rightarrow\infty}\frac{\log \sigma_0(n)}{\log n/\log\log n}=\log 2,\end{align*}

while

\begin{align*}\sum_{n\le N}\sigma_0(n)=N\log N + O(N).\end{align*}

In other words, $\sigma_0(n)$ behaves on average like $\log n$ , but it can also sometimes be larger than $2^{\xi\log n/\log\log n}$ , for any $\xi<1$ (see [ Reference Hardy and Wright10 , theorems 317, 320]).

As in Afifurraman’s result, our error term allows one to obtain significant estimates for X as small as $n^{1/2+\delta}$ . It turns out that this is the threshold of what one could hope for in the exponent, as demonstrated by our second result.

Theorem 2. For all prime numbers p we have that

\begin{equation*} S_{2,\|\cdot\|_\infty}(p,p^{1/2})=4\left(\frac{12}{\pi^2}-1\right)p+O\big(p^{1/2}\log p\big), \end{equation*}

and that

\begin{equation*} S_{2,\|\cdot\|_\infty}(p^2,p)=4\left(\frac{12}{\pi^2}-1\right)p^2+O\left(p\log p\right)\!.\end{equation*}

Note that the constant on the main term in these formulas is different from that in Theorem 1. This shows, in particular, that the main term in Theorem 1 does not give a correct asymptotic estimate, as $n\rightarrow\infty,$ with $X=n^{1/2}$ . As pointed out to us by one of the referees, this theorem is also related to the problem of counting directions determined by subsets of a finite field, as discussed in [ Reference Martin, White and Yip14 ]. In particular, Theorem 2 significantly improves the error term in [ Reference Martin, White and Yip14 , theorem 1·11] in the special case, in the statement of that theorem, when $\lambda=1$ .

Results similar to Theorem 2 for higher powers of primes do not seem to follow from the same types of arguments given in our proof. This is a direction for further investigation, which was the initial motivation for the research leading to the results in this paper, and which is closely related to a natural analogue of the ‘minimal denominator problem’ in the setting of p-adic numbers (see [ Reference Chen and Haynes5, Reference Marklof13, Reference Shparlinski20 ]).

Notation. We use O and $\ll$ to denote the standard big-oh and Vinogradov notations. If a and q are integers then we write (a, q) for the gcd of a and q. The symbols $\varphi,\mu,\omega,$ and $\zeta$ denote the Euler phi, Möbius, prime omega, and Riemann zeta functions, respectively. For $n\in\mathbb{N}$ , $\sum_{d|n}$ denotes a sum over positive divisors d of n. For $x\in\mathbb{R},$ we write $\lfloor x\rfloor$ for the greatest integer less than or equal to x and $\{x\}=x-\lfloor x\rfloor$ for the fractional part of x.