Distribution and moments of the error term in the lattice point counting problem for three-dimensional Cygan–Korányi balls

We study fluctuations of the error term for the number of integer lattice points lying inside a three-dimensional Cygan–Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

The problem of estimating ( ) arises naturally from the homogeneous structure imposed on the -th Heisenberg group H (when realized over R 2 +1 ), namely we have ( ) = Z 2 +1 ∩ B , where u = ( 1 , . . . , 2 , 2 2 +1 ) with > 0 are the Heisenberg dilations, and B = u ∈ R 2 +1 : |u| ≤ 1 is the unit ball with respect to the Cygan-Korányi norm (see [8,10] for more details). It is clear that ( ) will grow for large like vol B 2 +2 , where vol(·) is the Euclidean volume, and we shall be interested in the error term resulting from this approximation.
We shall refer to the lattice point counting problem for (2 + 1)-dimensional Cygan-Korányi balls as the problem of determining the value of . This problem was first considered by Garg, Nevo and Taylor [8], who established the lower bound ≥ 2 for all integers ≥ 1. For = 1, which is the case we shall be concerned with in the present paper, this lower bound was proven by the author to be sharp, that is 1 = 2 (see [9], Theorem 1.1, and note that a different normalization is used for the exponent of the error term). Thus, the lattice point counting problem for 3-dimensional Cygan-Korányi balls is settled. The behavior of the error term E ( ) in the lattice point counting problem for (2 +1)dimensional Cygan-Korányi balls with > 1 is of an entirely different nature compared to the case = 1. In the higher dimensional case ≥ 3, the best result available to date is |E ( )| ≪ 2 −2/3 which was proved by the author ( [10], Theorem 1), and we also have ( [10], Theorem 3) the Ω-result E ( ) = Ω 2 −1 log 1/4 log log 1/8 . It follows that 8 3 ≤ ≤ 3. In regards to what should be the conjectural value of in the case of ≥ 3, it is known ( [10], Theorem 2) that E ( ) has order of magnitude 2 −1 in mean-square, which leads to the conjecture that = 3. As for the case = 2, the conjectural value of 2 should be the same as in the higher dimensional case, namely 2 = 3. Our goal in the present paper is to investigate the nature in which 1 ( ) fluctuates around its expected value vol B 4 . In the higher dimensional case ≥ 3, this has been carried out by the author, and we have the following result ( [11], Theorem 1 and Theorem 3).
Theorem ( [11]). Let ≥ 3 be an integer, and let E ( ) = E ( )/ 2 −1 be the suitably normalized error term. Then there exists a probability density P ( ) such that, for any (piecewise)-continuous function F satisfying the growth condition |F ( )| ≪ 2 , we have The density P ( ) can be extended to the whole complex plane C as an entire function of , which satisfies for any non-negative integer ≥ 0, and any ∈ R, | | sufficiently large in terms of and , the decay estimate where > 0 is an absolute constant.
We are going to establish an analogue of the above result in the case of = 1, where the suitable normalization of the error term is given by E 1 ( ) = E 1 ( )/ 2 . We shall see once more a distinction between the case = 1 and the higher dimensional case ≥ 3, this time with respect to the probability density P ( ).
1.2. Statement of the main results.
Theorem 1. Let E 1 ( ) = E 1 ( )/ 2 be the suitably normalized error term. Then E 1 ( ) has a limiting value distribution in the sense that, there exists a probability density P 1 ( ) such that, for any (piecewise)-continuous function F of polynomial growth we have The density P 1 ( ) satisfies for any non-negative integer ≥ 0, and any ∈ R, | | sufficiently large in terms of , the decay estimate where > 0 is an absolute constant.
Remark. In the particular case where F ( ) = with ≥ 1 an integer, we have thereby establishing the existence of all moments (see the remark following Proposition 4 in §5 regarding quantitative estimates).
Remark. Note that the decay estimates (1.2) for the probability density in the case where = 1 are much stronger compared to the corresponding ones in the higher dimensional case ≥ 3 as stated in the introduction. Also, note that whereas for ≥ 3, the density P ( ) extends to the whole complex plane as an entire function of , and in particular is supported on all of the real line, Theorem 1 above makes no such claim in the case of = 1.
Our next result gives a closed form expression for all the integral moments of the density P 1 ( ). Theorem 2. Let ≥ 1 be an integer. Then the -th integral moment of P 1 ( ) is given by where the series on the RHS of (1.4) converges absolutely. For integers , ℓ ≥ 1, the term Ξ( , ℓ) is given by where (·) is the Möbius function, and 2 (·) is the counting function for the number of representation of an integer as the sum of two squares. For ℓ = 1 the sum in (1.4) is void, so by definition Ξ( , 1) = 0. In particular, it follows that Remark. There is a further distinction between the case = 1, and the higher dimensional case ≥ 3 in the following aspect. For = 1, P 1 ( ) is the probability density corresponding to the random series ∞ =1 ϕ 1, , where the are independent random variables uniformly distributed on the segment [0, 1], and the ϕ 1, ( ) are real-valued continuous functions, periodic of period 1, given by (see §3) This is in sharp contrast to the higher dimensional case ≥ 3, where the corresponding functions ϕ , ( ) (see [11], §2 Theorem 4) are aperiodic. Also, the presence of the factor 1/ in ϕ 1, ( ), as apposed to 1/ 3/4 in ϕ , ( ) for ≥ 3, is the reason for why we obtain the much stronger decay estimates (1.2) compared to the higher dimensional case ≥ 3.
Notation and conventions. The following notation will occur repeatedly throughout this paper. We use the Vinogradov asymptotic notation ≪, as well as the Bignotation. For positive quantities , > 0 we write ≍ , to mean that ≪ ≪ . In addition, we define (2) In this section, we develop a Voronoï-type series expansion for E 1 ( )/ 2 . The main result we shall set out to prove is the following.
Proposition 1. Let > 0 be large. Then for ≤ ≤ 2 we have for any > 0, where for > 0, ( ) is given by Remark. It is not difficult to show that −2 2 ≪ − for some > 0, and so one may replace this term by this bound which simplifies (2.1). However, we have chosen to retain the term −2 2 as we are going to show later on (see §3.1) that its average order is much smaller.
The proof of Proposition 1 will be given in §2.2. We shall first need to establish several results regarding weighted integer lattice points in Euclidean circles.
2.1. Weighted integer lattice points in Euclidean circles. We begin this subsection by proving Lemma 1 stated below, which will then be combined with Lemma 2 to prove Proposition 1. Then , for any > 0, where as any real number which satisfies ≍ .
Suppose first that > . We have (2.11) Moving the line of integration to ℜ( ) = − 1 4 , and using (2.11), we obtain By Stirling's asymptotic formula (2.4) it follows that where ( ) = −2 log + 2 + log 2 . Trivial integration and integration by parts give (2.14) Inserting (2.14) into (2.13), and then summing over all > , we obtain Inserting (2.15) into (2.9), we arrive at (2.17) Moving the line of integration to ℜ( ) = 1 − , and using (2.17), we obtain Extending the integral all the way to ±∞, by Stirling's asymptotic formula (2.4) we have where ( ) is defined as before. Trivial integration and integration by parts give Inserting (2.19) into the RHS of (2.18), we obtain by (2.20) Moving the line of integration in (2.21) to ℜ( ) = + 1/2 with ≥ 1 an integer, and then letting → ∞, we have by the theorem of residues where for > 0, the Bessel function J of order is defined by We have the following asymptotic estimate for the Bessel function (see [17], B.4 (B.35)). For fixed > 0, Inserting ( Summing over all ≤ , we obtain Finally, inserting (2.25) into the RHS of (2.16), we arrive at . This concludes the proof.
We need an additional result regarding weighted integer lattice points in Euclidean circles. First, we make the following definition.
Definition. For > 0 and ≥ 1 an integer, define We quote the following result (see [9], Lemma 2.1). Here, we shall only need that part of the lemma which concerns the case ≥ 1. The case = 0 will be treated by Lemma 1 as we shall see later.
Lemma 2 ([9]). For ≥ 1 an integer, the error term ( ) satisfies the bound where the implied constant is absolute.
Remark. The proof of Lemma 2 goes along the same line as the proof of Lemma 1, where in fact the proof is much simpler in this case. Moreover, one can show that the upper bound estimate (2.27) is sharp for = 1. For ≥ 2, the estimates are no longer sharp, but they will more then suffice for our needs.

2.2.
A decomposition identity for 1 ( ) and proof of Proposition 1. We have everything we need for the proof of Proposition 1. Before presenting the proof, we need the following decomposition identity for 1 ( ) which we prove in Lemma 3 below.
where 2 is defined as in Proposition 1.
Proof. The first step is to execute the lattice point count as follows. By the definition of the Cygan-Korányi norm, we have (2.29) Next, we decompose the first sum using the following procedure. For 0 ≤ ≤ 2 an integer, we use Taylor expansion to write Multiplying the above identity by 2 ( ), and then summing over the (2.30) Inserting (2.30) into the RHS of (2.29), we obtain This concludes the proof.
We now turn to the proof of Proposition 1.

Proof. (Proposition 1). Let
> 0 be large, and suppose that < < 2 . By the upper bound estimate (2.27) in Lemma 2, it follows from Lemma 3 that , where the infinite sum clearly converges absolutely. By the definition of 0 ( ), it is easily verified that Inserting (2.33) into the RHS of (2.32), we find that We claim that 2 = vol B . To see this, first note that by (2.2) in Lemma 1 we have the bound | 2 | ≪ log , and since | 2 | ≪ 2 , we obtain by (2.34) that Subtracting vol B 4 from both sides of (2.34), and then dividing throughout by 2 , we have by (2.2) in Lemma 1 upon choosing = 2 This concludes the proof of Proposition 1.

Almost periodicity
In this section we show that the suitably normalized error term E 1 ( )/ 2 can be approximated, in a suitable sense, by means of certain oscillating series. From this point onward, we shall use the notation E 1 ( ) = E 1 ( )/ 2 . The main result we shall set out to prove is the following.
Proposition 2. We have where ϕ 1,1 ( ), ϕ 1,2 ( ), . . . are real-valued continuous functions, periodic of period 1, given by The proof of Proposition 2 will be given in §3.2. Our first task will be to deal with the remainder term −2 2 appearing in the approximate expression (2.1).

3.1.
Bounding the remainder term. This subsection is devoted to proving the following lemma.
Lemma 4. We have Before commencing with the proof, we need the following result on trigonometric approximation for the function (see [27]).
Proof. (Lemma 4). Let > 0 be large. By Vaaler's Lemma with = , we have for in the range < < 2 Applying Cauchy-Schwarz inequality we obtain where for > 0 and ℎ ≥ 1 an integer, ℎ ( ) is given by Fix an integer 1 ≤ ℎ ≤ . Making a change of variable, we have where I ℎ ( , ) is given by We have the estimate Inserting (3.7) into the RHS of (3.5), we obtain (3.8) Integrating both sides of (3.4) and using (3.8), we find that This concludes the proof.
We end this subsection by quoting the following result (see [22]) which will be needed in subsequent sections of the paper.
Hilbert's inequality ( [22]). Let ( ) ∈Λ and ( ) ∈Λ be two sequences of complex numbers indexed by a finite set Λ of real numbers. Then Proof. (Proposition 2). Fix an integer ≥ 1, and let > 1/2 be large. In the range < < 2 , we have by Proposition 1 where , 2 ( ) is given by Integrating both sides of (3.10), we have by Lemma 4 and Cauchy-Schwarz inequality It remains to estimate the first term appearing on the RHS of (3.12). We have where for > 0 and ≥ 1 an integer, we define a ( ) = a( ) exp 2 √ . We first estimate the off-diagonal terms. By Hilbert's inequality, we have for = , 2 (3.14) Inserting (3.14) into the RHS of (3.13) and recalling that > 1/2 , it follows that (3.15) Inserting (3.15) into the RHS of (3.12), applying Cauchy-Schwarz inequality and then taking lim sup, we arrive at lim sup Finally, letting → ∞ in (3.16) concludes the proof.

The probability density
Having proved Proposition 2 in the last section, in this section we turn to the construction of the probability density P 1 ( ). We begin by making the following definition.
Definition. For ∈ C and ≥ 1 an integer, define and let We quote the following result (see [14], Lemma 2) which will be needed in subsequent sections of the paper.
We now state the main result of this section. More over, if we let  (II) For ∈ R, let P 1 ( ) = Φ 1 ( ) be the Fourier transform of Φ 1 . Then P 1 ( ) defines a probability density which satisfies for any non-negative integer ≥ 0 and any ∈ R, | | sufficiently large in terms of , the bound Proof. We begin with the proof of (I). Let ∈ C and ≥ 1 an integer. We are going apply Lemma 5 with b ( ) = exp 2 ϕ 1, ( ) , and frequencies = √ . The elements of the set B = { √ : | ( )| = 1} are are linearly independent over Q, and since ϕ 1, ( ) ≡ 0 whenever √ ∉ B (i.e. whenever is not square-free), we see that the conditions of Lemma 5 are satisfied, and thus (4.2) holds. Now, let us show that Φ 1 ( ) defines an entire function of . First we note the following. By the definition of ϕ 1, ( ), we have and we also have the uniform bound for some absolute constant > 0. By (4.5) and (4.6) we obtain whenever, say, ≥ | | 2 . Since ∞ =1 2 2 ( )/ 2 < ∞, it follows from (4.7) that the infinite product ∞ =1 Φ 1, ( ) converges absolutely and uniformly on any compact subset of the plane, and so Φ 1 ( ) defines an entire function of . We now estimate Φ 1 ( ) for large | |, = + . Let > 0 be an absolute constant such that (see [33]) and for real > e we write ( ) = 1 − /log log . Let > 0 be a small absolute constant which will be specified later, and set We are going to estimate the infinite product ∞ =1 Φ 1, ( ) separately for < ℓ and ≥ ℓ. In what follows, we assume that | | is sufficiently large in terms of and . The product over < ℓ is estimated trivially by using the upper bound (4.6), leading to for some absolute constant > 0. Suppose now that ≥ ℓ. By (4.8) we have It follows from (4.5), (4.6) and (4.10) that where the remainder term R ( ) satisfies the bound At this point we specify , by choosing 0 < < 1 64 such that 2 1/2 exp 2 ≤ 1 .
We now turn to the proof of (II). By the definition of the Fourier transform, we have for real and it follows from the decay estimates (4.23) that P 1 ( ) is of class ∞ . Let us estimate |P ( ) 1 ( )| for real , | | sufficiently large in terms . Let = , | | large, be a real number depending to be determined later, which satisfies sgn( ) = −sgn( ). By Cauchy's theorem and the decay estimate (4.23), we have (4.25) It follows that We decompose the range of integration in (4.26) as follows In what follows, we assume that | | is sufficiently large in terms of . In the range | | ≤ √ 2, we have by (4.23) From (4.28), it follows that L 1 satisfies the bound (4.29) Referring to (4.23) once again, we have in the range | | > √ 2 (4.30) where˜ is given by˜ We decompose the range of integration in (4.31) as follows In the range | | ≤ we estimate trivially, obtaining (4.33) In the range | | > we have | | Combining (4.33) and (4.34), we see that L 2 satisfies the bound (4.35) From (4.29) and (4.35) we find that L 1 dominates. By (4.26) and (4.27) we arrive at Finally, we specify . We choose With this choice, we have the bound (recall that | | is assumed to be large in terms of ) It remains to show that P 1 ( ) defines a probability density. This will be a consequence of the proof of Theorem 1. This concludes the proof of Proposition 3.

Power moment estimates
Having constructed the probability density P 1 ( ) in the previous section, our final task, before turning to the proof of the main results of this paper, is to establish the existence of all moments of the normalized error term E 1 ( ). The main result we shall set out to prove is the following.
Proposition 4. Let ≥ 1 be an integer. Then the -th power moment of E 1 ( ) is given by where the series on the RHS of (5.1) converges absolutely, and for integers , ℓ ≥ 1, the term Ξ( , ℓ) is given as in (1.4).
Remark. As we shall see later on, the RHS of (5.1) is simply ∫ ∞ −∞ P 1 ( )d . Also, our proof of Proposition 4 in fact yields (5.1) in a quantitative form, namely, we with an explicit decay estimate for the remainder term R ( ). However, as our sole focus here is on establishing the existence of the limit given in the LHS of (5.1), Proposition 4 will suffice for our needs.
It follows that say. Inserting (5.5) into the RHS of (5.4), we obtain (5.6) To estimate the off-diagonal terms, we use the identity sin = 1 2 exp ( ) − exp (− ) and then apply Hilbert's inequality, obtaining Inserting (5.7) into the RHS of (5.6), we arrive at Recalling that Ξ( , 1) = 0, it follows that the RHS of (5.1) in the case where = 2 is given by This proves (5.1) in the case where = 2.
In what follows, all implied constants in the Big notation are allowed to depend on . By Proposition 1 and Lemma 4, together with the trivial bound | −2 2 | ≪ 1, we have . Let 1 ≤ ≤ 2 be a large parameter to be determined later. In the range < < 2 , we have We first estimate the mean-square of the second summand appearing on the RHS of (5.11). We have (5.12) Applying Hilbert's inequality, we find that Inserting (5.13) into the RHS of (5.12), we obtain (5.14) Integrating both sides of (5.11), we have by (5.10), (5.14) and Cauchy-Schwarz inequality Before we proceed to evaluate the RHS of (5.15), we need the following result (see [32], §2 Lemma 2.2). Let e 1 , . . . , e = ±1, and suppose that 1 , . . . , ≤ are integers. Then it holds where the implied constant depends only . It follows from (5.16) that The estimate (5.17) in the case where =1 e √ ≠ 0 is somewhat wasteful, but nevertheless it will suffice for us. Inserting (5.17) into the RHS of (5.15), we obtain It remains to estimate the first summand appearing on the RHS of (5.18). Let e 1 , . . . , e = ±1. We have (5.20) where for ≥ 1, and integers , ℓ ≥ 1, the term Ξ( , ℓ; ) is given Denoting by (·) the divisor function, we have for square-free Using (5.22) repeatedly, it follows that where Ξ( , ℓ) is given as in (1.4). Using (5.23) repeatedly, and noting that Ξ( , 1; ) = 0, we have by (5.20) where the series appearing on the RHS of (5.24) converges absolutely. Finally, inserting (5.24) into the RHS of (5.18) and making the choice = 2 2− , we arrive at Collecting the results from the previous sections, we are now in a position to present the proof of the main results. We begin with the proof of Theorem 1.
Proof. (Theorem 1). We shall first prove (1.1) in the particular case where F ∈ ∞ 0 (R), that is, F is an infinitely differentiable function having compact support. To that end, let F be test function as above, and note that the assumptions on F imply that |F ( ) − F ( )| ≤ F | − | for all , , where F > 0 is some constant which depends on F . It follows that for any integer ≥ 1 and any > 0, we have where ϕ 1, ( ) is defined as in Proposition 2, and the remainder term E F , satisfies the bound By Proposition 2 it follows that Denoting by F the Fourier transform of F , the assumptions on the test function F allows us to write where ( ; ) is defined at the beginning of §4. Letting → ∞ in (6.4), we have by (4.2) in Proposition 3 together with an application of Lebesgue's dominated convergence theorem where ( ; ) = ≤ Φ 1, ( ), and Φ 1, ( ) is defined at the beginning of §4. Letting → ∞ in (6.5), and recalling that where in the second equality we made use of Parseval's theorem which is justified by the decay estimate (4.4) for P ( ) 1 ( ) with real stated in Proposition 3. It follows from (6.4), (6.5) and (6.6) that where the remainder term E ♭ F , satisfies Inserting (6.7) into the RHS of (6.1), we deduce from (6.3) and (6.8) that lim sup We conclude that (6.10) lim whenever F ∈ ∞ 0 (R). The result (6.10) extends easily to include the class 0 (R) of continuous functions with compact support. To see this, fix a smooth bump function ( ) ≥ 0 supported in [−1, 1] having total mass 1, and for an integer ≥ 1 let ( ) = ( ). Given F ∈ 0 (R), let F = F ★ ∈ ∞ 0 (R), where ★ denotes the Euclidean convolution operator. We then have it follows that (6.10) holds whenever F ∈ 0 (R).
Let us now consider the general case in which F is a continuous function of polynomial growth, say |F ( )| ≪ | | for all sufficiently large | |, where ≥ 1 is some integer. Let ∈ ∞ 0 (R) satisfy 0 ≤ ( ) ≤ 1, ( ) = 1 for | | ≤ 1, and set ( ) = ( / ). Define F ( ) = F ( ) ( ) ∈ 0 (R). For sufficiently large we have by Proposition 4 where the implied constant depends only on and the implicit constant appearing in the relation |F ( )| ≪ | | . Since F → F pointwise as → ∞, it follows from the rapid decay of P 1 ( ) that (6.14) We conclude from (6.13) and (6.14) that 1 ∫ 2 F E 1 ( ) d → ∫ ∞ −∞ F ( )P 1 ( )d as → ∞. It follows that (6.10) holds for all continuous functions of polynomial growth. The extension to include the class of (piecewise)-continuous functions of polynomial growth is now straightforward, and so (1.1) is proved. The decay estimates (1.2) stated in Theorem 1 have already been proved in Proposition 3, and it remains to show that P 1 ( ) defines a probability density. To that end, we note that the LHS of (6.10) is real and non-negative whenever F is. Since P 1 ( ) is continuous, by choosing a suitable test function F in (6.10), we conclude that P ( ) ≥ 0 for real . Taking F ≡ 1 in (6.10) we find ∫ ∞ −∞ P 1 ( )d = 1. The proof of Theorem 1 is therefor complete.
The proof of Theorem 2 is now immediate.