Abstract
Let \(Q({\mathbf{x}}) = Q(x_{1} ,x_{2} ,\ldots,x_{n} )\) be a nonsingular quadratic form with integer coefficients, n be even and p be an odd prime. In Hakami (J. Inequal. Appl. 2014:290, 2014, doi:10.1186/1029-242X-2014-290) we obtained an upper bound on the number of integer solutions of the congruence \(Q({\mathbf{x}}) \equiv 0\ (\operatorname{mod} p^{2} )\) in small boxes of the type \(\{ { {{\mathbf{x}} \in \mathbb{Z}_{p^{2} }^{n} | {a_{i} \leqslant x_{i} < a_{i} + m_{i} , 1 \leqslant i \leqslant n} } }\} \), centered about the origin, where \(a_{i} ,m_{i} \in\mathbb{Z}\), \(0< m_{i}\le p^{2}\), \(1 \leqslant i \leqslant n\). In this paper, we shall drop the hypothesis of ‘centered about the origin’ and generalize the result of paper Hakami (J. Inequal. Appl. 2014:290, 2014, doi:10.1186/1029-242X-2014-290) to boxes of arbitrary size and position.
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1 Introduction
Let \(Q({\mathbf{x}}) = Q(x_{1} ,x_{2} ,\ldots,x_{n} ) = \sum_{1 \leqslant i \leqslant j \leqslant n} a_{ij} x_{i} x_{j} \) be a quadratic form with integer coefficients in n-variables, p be an odd prime, \(\mathbb{Z}_{p^{2}}=\mathbb{Z}/(p^{2})\), and \(V_{p^{2}} = V_{p^{2} }(Q)\) be the algebraic subset of \(\mathbb{Z}_{p^{2} }^{n} \) defined by the equation
When n is even, we let \(\Delta_{p} (Q) = ( {( - 1)^{n/2} \det A_{Q} /p} )\) if \(p\nmid\det A_{Q} \) and \(\Delta_{p} (Q) = 0\) if \(p|\det A_{Q} \), where \((\cdot/p)\) denotes the Legendre-Jacobi symbol and \(A_{Q} \) is the \(n \times n\) defining matrix for \(Q({\mathbf{x}})\). We call Q a nonsingular form \((\operatorname{mod} p)\) if \(p \nmid\det A_{Q}\). As usual, we let \(|S|\) denote the cardinality of a set S.
Our first interest in this paper is obtaining an estimate for the number of solutions of (1.1) in a box of the type
viewed as a subset of \(\mathbb{Z}_{p^{2}}^{n}\), where \(a_{i} ,m_{i} \in\mathbb {Z}\), \(0< m_{i} \le p^{2}\), \(1 \le i \le n\).
Theorem 1
Suppose that n is even, Q is a nonsingular form \((\operatorname{mod} p)\) and that \(V_{p^{2}}(Q)\) is the set of solutions of (1.1). Then, for any box ℬ of type (1.2) (viewed as a subset of \(\mathbb{Z}_{p^{2}}^{n}\)) with \(0 < m_{i} \le p^{2}\), \(1 \le i \le n\), we have
where
We conjecture that the following upper bound holds:
which would be the best possible estimate. Indeed, for the form \(Q(\mathbf{x})=x_{1}x_{2}-x_{3}x_{4}\), the ϵ factor cannot be removed altogether. For this form it is known [1], Theorem 3, that the number of solutions of the equation \(Q(\mathbf{x})=0\) in integers x with \(1 \le x_{i} \le B\) is asymptotic to \(\frac{12}{\pi^{2}}B^{2} \log B\). Thus, for any B, the number of solutions of the congruence \(Q(\mathbf{x}) \equiv0 \ (\operatorname{mod}p^{2})\) with \(1 \le x_{i} \le B\) is at least \(\frac{12}{\pi ^{2}}B^{2} \log B\). Letting \(B \approx p\) demonstrates the optimality of the conjectured upper bound. In Section 3 we establish the asymptotic estimate
The error term \(p^{n}\) in the upper bound (1.3) greatly improves on the error term \(p^{\frac{3}{2}n -1}\log^{n} p\) in the asymptotic estimate at the expense of having to place a constant larger than 1 on the main term. We would expect that the error term in the asymptotic estimate can be improved at least to the value \(p^{n}\) appearing in our upper bound.
In the next theorem the same type of bound as Theorem 1 is given for boxes with sides of unrestricted lengths. In this case, we let \({V_{{p^{2}},\mathbb{Z}}}\) denote the set of integer solutions of the congruence
and regard ℬ as a set of points in \(\mathbb{Z}^{n}\).
Theorem 2
Suppose that n is even, Q is nonsingular \((\operatorname{mod} p)\) and \(V_{p^{2} ,\mathbb{Z}} = V_{p^{2} ,\mathbb{Z}} (Q)\) is the set of integer solutions of the congruence (1.5). Then, for any box ℬ of type (1.2) (allowing \(m_{i}>p^{2}\)), we have
where \(\gamma_{n}\) is as in (1.4), and
We devote Section 4 and Section 5 respectively to the proofs of Theorem 1 and Theorem 2.
2 Preliminary lemmas
For any x, y in \(\mathbb{Z}_{p^{2} }^{n} \), we let \({\mathbf{x}} \cdot{\mathbf{y}}\) denote the ordinary dot product \({\mathbf{x}} \cdot{\mathbf{y}} = \sum_{i = 1}^{n} x_{i} y_{i} \). For any \(x \in\mathbb{Z}_{p^{2} } \), let \(e_{p^{2} } (x) = e^{2\pi ix/p^{2} } \). We use the abbreviation \(\sum_{\mathbf{x}} = \sum_{{\mathbf{x}} \in\mathbb{Z}_{p^{2} }^{n} } \) for complete sums. For \(\mathbf{y}\in\mathbb{Z}_{p^{2}}^{n}\), we write \(p|\mathbf{y}\) if \(p|y_{i}\), \(1 \le i \le n\) (where the \(y_{i}\) are regarded as integer representatives for the residue classes). In this case \(\frac{1}{p} \mathbf{y}\) is a well-defined element of \(\mathbb{Z}_{p^{2}}^{n}\). Let Q be a nonsingular quadratic form \((\operatorname{mod} p)\), and \(V_{p^{2}}=V_{p^{2}}(Q)\) be the set of solutions of (1.1). For \(\mathbf{y} \in\mathbf{Z}_{p^{2}}^{n}\) we define
The following lemma was established in [2].
Lemma 1
Suppose that n is even, Q is nonsingular modulo p and \(\Delta = \Delta_{p} (Q)\). Then, for any \({\mathbf{y}}\in\mathbb{Z}_{p^{2}}^{n}\),
where \(Q^{*} \) is the quadratic form associated with the inverse of the matrix for Q modp.
In [3] we established the basic identity
for any complex valued function \(\alpha(\mathbf{x})\) defined on \(\mathbb{Z}_{p^{2}}\) with Fourier expansion
Inserting the value of \(\phi(V_{p^{2}},{\mathbf{y}})\) from Lemma 1 into the basic identity (2.1) yields the following (see [4]).
Lemma 2
(The fundamental identity)
For any complex valued \(\alpha ({\mathbf{x}})\) on \(\mathbb{Z}_{p^{2} }^{n} \),
3 Asymptotic estimate of \(|\mathcal{B} \cap V_{p^{2}}|\)
To obtain an asymptotic estimate for the number of solutions of (1.5) in a box ℬ with sides of length \(m_{i} \le p^{2}\), we let \(\alpha= \chi_{\mathcal{B}}\), the characteristic function for the box. For such α, it is well known that the Fourier coefficients \(a_{\mathcal{B}}( \mathbf{y})\) have magnitude
where the term in the product is taken to be \(m_{i} \) if \(y_{i} = 0\). Henceforth, we choose representatives y for \(\mathbb{Z}_{p^{2}}^{n}\) with \(-\frac{p^{2}-1}{2} \le y_{i} \le\frac{p^{2}-1}{2}\), \(1 \le i \le n\). With this convention we can say
from which one readily obtains the well-known inequality
Also, by Lemma 1 one has uniformly \(|\phi(V_{p^{2}},\mathbf{y})| \le p^{\frac{3}{2} n-1}+p^{n}\). The asymptotic formula in (1.3) is now an immediate consequence of the basic identity (2.1), and the fact that \(a_{\mathcal{B}}(\mathbf{0}) = |\mathcal{B}|/p^{2n}\).
4 Proof of Theorem 1
We turn now to the proof of Theorem 1. Let ℬ be a box of point of the type (1.2), with \(0 < m_{i} \le p^{2}\), \(1 \le i \le n\), and let \(\chi_{\mathcal{B}} \) be its characteristic function with Fourier expansion
As usual, we define the convolution of two functions α, β defined on \(\mathbb{Z}_{p^{2}}\) by
Lemma 3
Let \(\alpha = \chi_{\mathcal{B}} * \chi_{\mathcal{B}'}\), where ℬ is a box as in (1.2), \(\mathcal{B}' = \mathcal{B} - {\mathbf{c}}\), with c chosen so that \(\mathcal{B}'\) is ‘nearly’ centered at the origin,
Then, for any subset S of \(\mathbb{Z}_{p^{2}}^{n}\), we have
Proof
Let
Then if \(m_{i}\) is odd, \(c_{i} = a_{i} + \frac{{m_{i} - 1}}{2}\), and hence
Thus, for any \(x \in I\),
If \(m_{i}\) is even, so that \(c_{i} = a_{i} + \frac{{m_{i} }}{2} - 1\), then
and so for any \(x \in I\),
Thus, for any \(\mathbf{x} \in\mathcal{B}\), we have
and so for any subset S of \(\mathbb{Z}_{p^{2}}^{n}\),
□
With α as given in Lemma 3, we have by the fundamental identity, Lemma 2, that
Also,
and
It follows that
By the Cauchy-Schwarz inequality and Parseval’s identity (see, for example, [5, 6]), we get
Next
where
Continuing from (4.3) and using (4.2), we obtain
Also,
where
Continuing from (4.5),
We are left with estimating \(\sum_{\vert {y_{i} } \vert < p/2} \vert {a_{i} (py_{i} )} \vert \). Say \(a({\mathbf{y}}) = \prod_{i = 1}^{n} a_{i} (y_{i} )\). Since the Fourier coefficients are given by \(a({\mathbf{y}}) = p^{2n} a_{B} ({\mathbf{y}}) a_{B'} ({\mathbf{y}})\), we have
and so
Lemma 4
Proof
We begin by establishing the inequality
We split the proof of the inequality into two cases.
Case (I): If \(\frac{p}{{2m_{i} }} \geqslant1\), then
Thus,
and so
Case (II): If \(\frac{p}{{2m_{i} }} < 1\), then
Returning to the proof of the lemma, we consider four cases as follows.
Case (i): If \(m_{i} \leqslant\frac{p}{ 2}\), then by (4.7) and (4.8) we have
Case (ii): If \(m_{i} > \frac{p}{2}\), then by (4.7) and (4.8)
Case (iii): If \(\frac{p}{2} < m_{i} < p\), then continuing from Case (ii) we have
Case (iv): If \(m_{i} > p\), then continuing from Case (ii) we get
completing the proof of Lemma 4. □
We return to the proof of Theorem 1. Suppose that
By Lemma 4, we obtain
Using (4.9), then continuing from (4.6), we have
By (4.1) and (4.4), we then obtain
The task now is to determine which of the terms \(\vert \mathcal{B} \vert ^{2} /p^{2} \), \(p^{n} \vert \mathcal{B} \vert \) and \(3^{n} 2^{l} p^{l - (n/2) - 1} \vert \mathcal{B} \vert \prod_{i = l + 1}^{n} {m_{i} } \) in (4.10) is the dominant term. We consider two cases as follows.
Case (i): Suppose \(l \leqslant\frac{n}{2} - 1\). Then, comparing the first and third terms, we get
This leads to
Case (ii): Suppose \(l \geqslant\frac{n}{2}\). Then, comparing the second and third terms, we have
This gives that
So for any l, we always have
Returning to (4.10), we now can write
where \(\gamma'_{n} = 1 + 3^{n} 2^{l} \). On the other hand, using Lemma 3, we have
Combining the last two inequalities ((4.11) and (4.12)) yields
where \(\gamma_{n}=1+6^{n}\). Theorem 1 is proved.
5 Proof of Theorem 2
Let ℬ be a box of points in \(\mathbb{Z}^{n}\) as given in (1.2). Partition ℬ into \(N = N_{\mathcal{B}} \) smaller boxes \(\mathrm{B}_{i} \),
where each \(\mathrm{B}_{i} \) has all of its edge lengths \(\leqslant p^{2} \). Plainly,
Applying Theorem 1 to each \(\mathrm{B}_{i} \), we get
The proof of Theorem 2 is complete.
References
Ayyad, A, Cochrane, T, Zheng, Z: The congruence \(x_{1}x_{2}\equiv x_{3}x_{4}\ (\operatorname{mod} p)\), the equation \(x_{1}x_{2}=x_{3}x_{4}\), and mean values of character sums. J. Number Theory 59(2), 398-413 (1996)
Hakami, A: Small zeros of quadratic congruences to a prime power modulus. PhD thesis, Kansas State University (2009)
Hakami, A: Estimates for lattice points of quadratic forms with integral coefficients modulo a prime number square. J. Inequal. Appl. 2014, 290 (2014). doi:10.1186/1029-242X-2014-290
Hakami, A, Cochrane, T: Small zeros of quadratic forms mod \(p^{2}\). Proc. Am. Math. Soc. 140(12), 4041-4052 (2012)
Cochrane, T: Small solutions of congruences. PhD thesis, University of Michigan (1984)
King, HL: Introduction to Number Theory. Springer, Berlin (1982)
Acknowledgements
The author would like to thank his professor Todd Cochrane for suggesting the idea behind the statement of Lemma 3, which inspired the results of this paper. He would also like to thank the referees for their detailed comments and suggestions which improved the presentation of the results of this paper. Finally, the author would like to thank the VTEX Typesetting Services for their assistance in formatting and typesetting this paper.
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Hakami, A.H. Estimates for lattice points of quadratic forms with integral coefficients modulo a prime number square (II). J Inequal Appl 2015, 110 (2015). https://doi.org/10.1186/s13660-015-0637-0
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DOI: https://doi.org/10.1186/s13660-015-0637-0