Statistics of Lattice Points

in Thin Annuli for Generic Lattices

Igor Wigman

Received: August 22, 2005

Communicated by Friedrich Götze

Abstract. We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the Gaussian distribution. If the width shrinks slowly to zero, the conjecture was proven by Hughes and Rudnick for the standard lattice, and in our previous paper for generic rectangular lattices. We prove this conjecture for arbitrary lattices satisfying some generic Diophantine properties, again assuming the width of the annuli shrinks slowly to zero. One of the obstacles of applying the technique of Hughes-Rudnick on this problem is the existence of so-called close pairs of lattice points. In order to overcome this difficulty, we bound the rate of occurence of this phenomenon by extending some of the work of Eskin-Margulis-Mozes on the quantitative Openheim conjecture.

2000 Mathematics Subject Classification: Primary: 11H06, Secondary: 11J25 Keywords and Phrases: Lattice, Counting Function, Circle, Ellipse, Annulus, Two-Dimensional Torus, Gaussian Distribution, Diophantine approximation

1 Introduction

We consider a variant of the circle problem. Let Λ ⊂ ℝ^{2} be a planar lattice, with det Λ the area of its fundamental cell. Let

N_{Λ}(t) = {x ∈ Λ : |x| ≤ t},

denote its counting function, that is, we are counting Λ-points inside a disc of radius t. As well known, as t → ∞, N_{Λ}(t) ∼ \frac{π}{det Λ}t^{2}. Denoting the remainder or the error term

Δ_{Λ}(t) = N_{Λ}(t) − \frac{π}{det Λ}t^{2},

it is a conjecture of Hardy that

|Δ_{Λ}(t)| ≪_{ϵ} t^{1/2+ϵ}.

Another problem one could study is the statistical behavior of the value distribution of Δ_{Λ} normalized by \sqrt{t}, namely of

F_{Λ}(t) := \frac{Δ_{Λ}(t)}{\sqrt{t}}.

Heath-Brown [HB] shows that for the standard lattice Λ = ℤ^{2}, the value distribution of F_{Λ}, weakly converges to a non-Gaussian distribution with density p(x). Bleher [BL3] established an analogue of this theorem for a more general setting, where in particular it implies a non-Gaussian limiting distribution of F_{Λ}, for any lattice Λ ⊂ ℤ^{2}. However, the object of our interest is slightly different. Rather than counting lattice points in the circle of varying radius t, we will do the same for annuli. More precisely, we define

N_{Λ}(t, ρ) := N_{Λ}(t + ρ) − N_{Λ}(t),

that is, the number of Λ-points inside the annulus of inner radius t and width ρ. The ”expected” value is the area \frac{π}{det Λ}(2tρ + ρ^{2}), and the corresponding normalized remainder term is

S_{Λ}(t, ρ) := \frac{N_{Λ}(t + ρ) − N_{Λ}(t) − \frac{π}{det Λ}(2tρ + ρ^{2})}{\sqrt{t}}.

The statistics of S_{Λ}(t, ρ) vary depending to the size of ρ(t). Of our particular interest is the intermediate or macroscopic regime. Here ρ → 0, but ρt → ∞. A particular case of the conjecture of Bleher and Lebowitz [BL4] states that S_{Λ}(t, ρ) has a Gaussian distribution. In 2004 Hughes and Rudnick [HR] established the Gaussian distribution for the unit circle, under an additional assumption that ρ(t) ≫ t^{−ϵ} for every ϵ > 0. By a rotation and dilation (which does not effect the counting function), we may assume, with no loss of generality, that Λ admits a basis one of whose elements is the vector (1, 0), that is Λ = ⟨1, α + iβ⟩ (we make the natural identification of i with (0, 1)). In a previous paper [W] we already dealt with the problem of investigating the statistical properties of the error term for rectangular lattice Λ = ⟨1, iβ⟩. We established the limiting Gaussian distribution for the ”generic” case in this 1-parameter family.

Some of the work done in [W] extends quite naturally for the 2-parameter family of planar lattices ⟨1, α + iβ⟩. That is, in the current work we will require the algebraic independence of α and β, as well as a strong Diophantine property of the \underline{pair} (α, β) (to be defined), rather than the transcendence and a strong Diophantine property of the aspect ratio of the ellipse, as in [W]. We say that a real number ξ is strongly Diophantine, if for every fixed natural

n, there exists K_{1} > 0, such that for integers a_{j} with ∑_{j=0}^{n} a_{j}ξ^{j} ≠ 0,

| ∑_{j=0}^{n} a_{j}ξ^{j}| ≫_{n} \frac{1}{( max_{0≤j≤n} |a_{j}|)^{K_{1}}} .

It was shown by Mahler [MAH], that this property holds for a ”generic” real number. We say that a pair of numbers (α, β) is strongly Diophantine, if for every fixed natural n, there exists a number K_{1} > 0, such that for every integral polynomial p(x, y) = ∑_{i+j≤n} a_{i, j}x^{i}y^{j} of degree ≤ n, we have

|p(α, β)| ≫_{n} \frac{1}{max_{i+j≤n} |a_{i, j}|^{K_{1}}} ,

whenever p(α, β) ≠ 0. This holds for almost all real pairs (α, β), see section 2.2.

Theorem 1.1. Let Λ = ⟨1, α + iβ⟩ where (α, β) is algebraically independent and strongly Diophantine pair of real numbers. Assume that ρ = ρ(T) → 0, but for every δ > 0, ρ ≫ T^{−δ}. Then for every interval A,

lim_{T →∞} \frac{1}{T} meas\{t ∈ [T, 2T] : \frac{S_{Λ}(t, ρ)}{σ} ∈ A\} = \frac{1}{\sqrt{2π}} ∫_{A} e^{− \frac{x^{2}}{2}} dx,  (1)

where the variance is given by

σ^{2} := \frac{4π}{β} · ρ.  (2)

Remark: Note that the variance σ^{2} is α-independent, since the determinant det(Λ) = β. One of the features of a rectangular lattice is that it is quite easy to show that the number of so-called close pairs of lattice points or pairs of points lying within a narrow annulus is bounded by essentially its average (see lemma 5.2 of [W]). This particular feature of the rectangular lattices was exploited while reducing the computation of the moments to the ones of a smooth counting function (we call it ”unsmoothing”). In order to prove an analogous bound for a general lattice, we extend a result from Eskin, Margulis and Mozes [EMM] for our needs to obtain proposition 3.1. We believe that this proposition is of independent interest.

2 The distribution of S̃_{Λ, M, L}

We apply the same smoothing as in [HR] and [W]: let χ be the indicator function of the unit disc and ψ a nonnegative, smooth, even function on the real line, of total mass unity, whose Fourier transform, ψ̂ is smooth and has compact support. Introduce a rotationally symmetric function Ψ on ℝ^{2} by setting Ψ̂(y⃗) = ψ̂(|y⃗|), where | · | denotes the standard Euclidian norm. For ϵ > 0, set

Ψ_{ϵ}(x⃗) = \frac{1}{ϵ^{2}} Ψ(\frac{x⃗}{ϵ}).

Define a smooth counting function

Ñ_{Λ,M}(t) = ∑_{n⃗∈Λ} χ_{ϵ}(\frac{n⃗}{t}),  (3)

with ϵ = ϵ(M) and χ_{ϵ} = χ ∗ Ψ_{ϵ}, the convolution of χ with Ψ_{ϵ}. In what will follow,

ϵ = \frac{1}{t\sqrt{M}} ,  (4)

where M = M(T) is the smoothing parameter, which tends to infinity with t. In this section, we are interested in the distribution of the smooth version of S_{Λ}(t, ρ), denoted S̃_{Λ, M, L}(t), where L := \frac{1}{ρ}, defined by

S̃_{Λ, M, L}(t) = \frac{Ñ_{Λ, M}(t + \frac{1}{L}) − Ñ_{Λ, M}(t) − \frac{π}{d} (\frac{2t}{L} + \frac{1}{L^{2}} )}{\sqrt{t}},  (5)

We assume that for every δ > 0, L = L(T) = O(T^{δ}), which corresponds to the assumption of theorem 1.1 regarding ρ := \frac{1}{L}. Rather than drawing t at random from [T, 2T] with a uniform distribution, we prefer to work with smooth densities: introduce ω ≥ 0, a smooth function of total mass unity, such that both ω and ω̂ are rapidly decaying, namely

|ω(t)| ≪ \frac{1}{(1 + |t|)^{A}} , |ω̂(t)| ≪ \frac{1}{(1 + |t|)^{A}} ,

for every A > 0. Define the averaging operator

⟨f⟩_{T} = \frac{1}{T} ∫_{−∞}^{∞} f(t)ω(\frac{t}{T} )dt,

and let ℙ_{ω, T} be the associated probability measure:

ℙ_{ω, T} (f ∈ A) = \frac{1}{T} ∫_{−∞}^{∞} 1_{A}(f(t))ω(\frac{t}{T} )dt.

Remark: In what follows, we will suppress the explicit dependency on T, whenever convenient.

Theorem 2.1. Suppose that M(T) and L(T) are increasing to infinity with T, such that M = O(T^{δ}) for all δ > 0, and L/\sqrt{M} → 0. Then if (α, β) is an algebraically independent strongly Diophantine pair, we have for Λ = ⟨1, α + iβ⟩,

lim_{T →∞} ℙ_{ω, T} \{\frac{S̃_{Λ, M, L}}{σ} ∈ A\} = \frac{1}{\sqrt{2π}} ∫_{A} e^{− \frac{x^{2}}{2}} dx,

for any interval A, where

σ^{2} := \frac{4π}{βL}.  (6)

Definition: A tuple of real numbers (α_{1}, …, α_{n}) ∈ ℝ^{n} is called Diophantine, if there exists a number K > 0, such that for every integer tuple {a_{i}}_{i=0}^{n},

|a_{0} + ∑_{i=1}^{n} a_{i}α_{i}| ≫ \frac{1}{q^{K}} ,  (7)

with q = max_{0≤i≤n} |a_{i}|, whenever the LHS of the inequality doesn’t vanish. Khint-

chine proved that almost all tuples in ℝ^{n} are Diophantine (see, e.g. [S], pages 60-63). Denote the dual lattice

Λ^{∗} = ⟨1, γ + iδ⟩

with γ = −\frac{α}{β} and δ = \frac{1}{β}. In the rest of the current section, we assume, that, unless specified otherwise, the set of the squared lengths of vectors in Λ^{∗} satisfy the Diophantine property. That means, that (α^{2}, αβ, β^{2}) is a Diophantine triple of real numbers. We may assume (α^{2}, αβ, β^{2}) being Diophantine, since theorem 1.1 (and theorem 2.1) assume (α, β) is strongly Diophantine, which is, obviously, a stronger assumption. We use the following approximation to Ñ_{Λ,M}(t) (see e.g [W], lemma 4.1), which holds unconditionally on any Diophantine assumption:

Lemma 2.2. As t → ∞,

Ñ_{Λ,M}(t) = \frac{πt^{2}}{β} − \frac{\sqrt{t}}{βπ} ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{cos (2πt|k⃗| + \frac{π}{4} )}{|k⃗|^{\frac{3}{2}}} · ψ̂(\frac{|k⃗|}{\sqrt{M}} ) + O(\frac{1}{\sqrt{t}}), (8)

where, again, Λ^{∗} is the dual lattice.

By the definition of S̃_{Λ, M, L} in (5) and appropriately manipulating the sum in (8) we obtain the following Corollary 2.3.

S̃_{Λ, M, L}(t) = \frac{2}{βπ} ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{sin (\frac{π|k⃗|}{L})}{|k⃗|^{\frac{3}{2}}} sin (2π(t + \frac{1}{2L})|k⃗| + \frac{π}{4} )ψ̂(\frac{|k⃗|}{\sqrt{M}} )

+ O(\frac{1}{\sqrt{t}}) .

(9)

One should note that ψ̂ being compactly supported means that the sum essentially truncates at |k⃗| ≈ \sqrt{M}. Unlike the standard lattice, clearly there are no nontrivial multiplicities in Λ, that is

Lemma 2.4. Let a⃗_{j} = m_{j} + n_{j}(α + iβ) ∈ Λ, j = 1, 2, with an irrational α such that β ∉ ℚ(α). Then if |a⃗_{1}| = |a⃗_{2}|, either n_{1} = n_{2} and m_{1} = m_{2} or n_{1} = −n_{2} and n_{2} = −m_{2}.

Proof of theorem 2.1. We will show that the moments of S̃_{Λ, M, L} corresponding to the smooth probability space converge to the moments of the normal distribution with zero mean and variance which is given by theorem 2.1. This allows us to deduce that the distribution of S̃_{Λ, M, L} converges to the normal distribution as T → ∞, precisely in the sense of theorem 2.1. First, we show that the mean is O(\frac{1}{\sqrt{T}}). Since ω is real,

|⟨ sin (2π(t + \frac{1}{2L})|k⃗| + \frac{π}{4} )⟩| = |ℑm\{ω̂( − T |k⃗|)e^{iπ( \frac{|k⃗|}{L} + \frac{1}{4}} \}| ≪ \frac{1}{T^{A}|k⃗|^{A}}

for any A > 0, where we have used the rapid decay of ω̂. Thus

|⟨S̃_{Λ, M, L}⟩| ≪ ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{1}{T^{A}|k⃗|^{A+3/2}} + O(\frac{1}{\sqrt{T}} ) ≪ O(\frac{1}{\sqrt{T}} ),

due to the convergence of ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{1}{|k⃗|^{A+3/2}} , for A > \frac{1}{2}

Now define

M_{Λ, m} := ⟨(\frac{2}{βπ} ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{sin (\frac{π|k⃗|}{L})}{|k⃗|^{\frac{3}{2}}} sin (2π(t + \frac{1}{2L})|k⃗| + \frac{π}{4} )ψ̂(\frac{|k⃗|}{\sqrt{M}} ))^{m}⟩

(10)

Then from (9), the binomial formula and the Cauchy-Schwartz inequality,

⟨(S̃_{Λ, M, L})^{m}⟩ = M_{Λ, m} + O( ∑_{j=1}^{m} \binom{m}{j}\frac{\sqrt{M_{2m−2j}}}{T^{j/2}} )

Proposition 2.5 together with proposition 2.8 allow us to deduce the result of theorem 2.1 for an algebraically independent strongly Diophantine (ξ, η) := (−\frac{α}{β}, \frac{1}{β}). Clearly, (α, β) being algebraically independent and strongly Diophantine is sufficient.

2.1 The variance

The computation of the variance is done in two steps. First, we reduce the main contribution to the diagonal terms, using the assumption on the pair (α, β) (i.e. (α^{2}, αβ, β^{2}) is Diophantine). Then we compute the contribution of the diagonal terms. Both these steps are very close to the corresponding ones in [W]. Suppose that the triple (α^{2}, αβ, β^{2}) satisfies (7).

Proposition 2.5. If M = O(T^{1/(K+1/2+δ)}) for fixed δ > 0, then the variance of S̃_{Λ, M, L} is asymptotic to

σ^{2} := \frac{4}{β^{2}π^{2}} ∑_{k⃗∈Λ^{∗}\\{0\}} \frac{sin^{2} (\frac{π|k⃗|}{L})}{|k⃗|^{3}} ψ̂^{2}(\frac{|k⃗|}{\sqrt{M}} )

If L → ∞, but L/\sqrt{M} → 0, then

σ^{2} ∼ \frac{4π}{βL}  (11)

Remark: In the formulation of proposition 2.5, K is implicitly given by (7).

Proof. Expanding out (10), we have

M_{Λ, 2} = \frac{4}{β^{2}π^{2}} ∑_{k⃗,l⃗∈Λ^{∗}\\{0\}} \frac{sin (\frac{π|k⃗|}{L}) sin (\frac{π|l⃗|}{L})ψ̂(\frac{|k⃗|}{\sqrt{M}} )ψ̂(\frac{|l⃗|}{\sqrt{M}} )}{|k⃗|^{\frac{3}{2}} |l⃗|^{\frac{3}{2}}}

× ⟨ sin (2π(t + \frac{1}{2L})|k⃗| + \frac{π}{4} ) sin (2π(t + \frac{1}{2L})|l⃗| + \frac{π}{4} )⟩

(12)

It is easy to check that the average of the second line of the previous equation is:

\begin{matrix}\frac{1}{4}[ & ω̂(T (|k⃗| − |l⃗|))e^{iπ(1/L)(|l⃗|−|k⃗|)}+ \\ ω̂(T (|l⃗| − |k⃗|))e^{iπ(1/L)(|k⃗|−|l⃗|)}+ \\ ω̂(T (|k⃗| + |l⃗|))e^{−iπ(1/2+(1/L)(|k⃗|+|l⃗|))}− \\ ω̂( − T (|k⃗| + |l⃗|))e^{iπ(1/2+(1/L)(|k⃗|+|l⃗|))}] \end{matrix}  (13)

Documenta Mathematica 11 (2006) 1–23

Recall that the support condition on ψ̂ means that k⃗ and l⃗ are both constrained to be of length O(\sqrt{M}). Thus the off-diagonal contribution (that is for |k⃗| ≠ |l⃗| ) of the first two lines of (13) is

≪ ∑_{\begin{matrix}k⃗,l⃗∈Λ^{∗}\\{0\} \\ |k⃗|, |k⃗′|≤\sqrt{M} \end{matrix}} \frac{M^{A(K+1/2)}}{T^{A}} ≪ \frac{M^{A(K+1/2)+2}}{T^{A}} ≪ T^{−B},

for every B > 0, since (α, αβ, β^{2}) is Diophantine. Obviously, the contribution to (12) of the two last lines of (13) is negligible both in the diagonal and off-diagonal cases, justifying the diagonal approximation of (12) in the first statement of the proposition. To compute the asymptotics, we write we take a large parameter Y = Y (T) > 0 (to be chosen later), and write:

∑_{k⃗∈Λ^{∗}\\{0\}} \frac{sin^{2} (\frac{π|k⃗|}{L})}{|k⃗|^{3}} ψ̂^{2}(\frac{|k⃗|}{\sqrt{M}} ) = ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|^{2}≤Y \end{matrix}} + ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|^{2}>Y \end{matrix}} := I_{1} + I_{2},

Now for Y = o(M), ψ̂^{2}(\frac{|k⃗|}{\sqrt{M}} ) ∼ 1 within the constraints of I_{1}, and so

I_{1} ∼ ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|^{2}≤Y \end{matrix}} \frac{sin^{2} (\frac{π|k⃗|}{L})}{|k⃗|^{3}}.

Recall that Λ^{∗} = ⟨1, γ + iδ⟩. The sum in

∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|^{2}≤Y \end{matrix}} \frac{sin^{2} (\frac{π|k⃗|}{L})}{|k⃗|^{3}} = \frac{1}{L} ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|^{2}≤Y \end{matrix}} \begin{matrix}sin & 2 & ( & \frac{π|k⃗|}{L} & ) \\ ( & \frac{|k⃗|}{L} & ) & 3 \end{matrix} \frac{1}{L^{2}} .

is a 2-dimensional Riemann sum of the integral

\begin{aligned}∫∫_{1/L^{2}≪(x+yγ)^{2}+(δy)^{2}≤Y/L^{2}} & \frac{sin^{2} (π\sqrt{(x + yγ)^{2} + (δy)^{2}})}{|(x + yγ)^{2} + (δy)^{2}|^{3/2}}dxdy \\ & ∼ \frac{2π}{δ} ∫_{\frac{1}{L}}^{\frac{\sqrt{Y}}{L}} \frac{sin^{2}(πr)}{r^{2}}dr → βπ^{3}, \end{aligned}

Documenta Mathematica 11 (2006) 1–23

provided that Y/L^{2} → ∞, since ∫_{0}^{∞} \frac{sin^{2}(πr)}{r^{2}}dr = \frac{π^{2}}{2} . We changed the coordinates

appropriately. And so,

I_{1} ∼ \frac{βπ^{3}}{L}

Next we will bound I_{2}. Since ψ̂ ≪ 1, we may use the same change of variables to obtain:

\begin{aligned}I_{2} & ≪ \frac{1}{L} ∫∫_{(x+yγ)^{2}+(δy)^{2}≥Y/L^{2}} \frac{sin^{2} (π\sqrt{(x + yγ)^{2} + (δy)^{2}})}{|(x + yγ)^{2} + (δy)^{2}|^{3/2}}dxdy \\ & ≪ \frac{1}{L} ∫_{\sqrt{Y} /L}^{∞} \frac{dr}{r^{2}} = o(\frac{1}{L}). \end{aligned}

This concludes the proposition, provided we have managed to choose Y with L^{2} = o(Y) and Y = o(M). Such a choice is possible by the assumption of the proposition regarding L.

2.2 The higher moments

In order to compute the higher moments we will prove that the main contribution comes from the so-called diagonal terms (to be explained later). Our bound for the contribution of the off-diagonal terms holds for a strongly Diophantine pair of real numbers, which is defined below. In order to show that the strongly Diophantine pairs are ”generic”, we use theorem 2.6 below, which is a consequence of the work of Kleinbock and Margulis [KM]. The contribution of the diagonal terms is computed exactly in the same manner it was done in [W], and so we will omit it here.

Definition: We call the pair (ξ, η) strongly Diophantine, if for all natural n there exists a number K_{1} = K_{1}(ξ, η, n) ∈ ℕ such that for every integral polynomial of 2 variables p(x, y) = ∑_{i+j≤n} a_{i, j}x^{i}y^{j} of degree ≤ n, we have

|p(ξ, η)| ≫ h^{−K_{1}},  (14)

where h = max_{i+j≤n} |a_{i, j}| is the height of p. The constant involved in the ” ≫ ”

notation may depend only on ξ, η, n and K_{1}.

Theorem 2.6. Let an integer n be given. Then almost all pairs of real numbers (ξ, η) ∈ ℝ^{2} satisfy the following property: there exists a number K_{1} = K_{1}(n) ∈ ℕ such that for every integer polynomial of 2 variables p(x, y) = ∑_{i+j≤n} a_{i, j}x^{i}y^{j}

of degree ≤ n, (14) is satisfied.

Theorem 2.6 states that almost all real pairs of numbers are strongly Diophantine.

Remark: Theorem A in [KM] is much more general than the result we are using. As a matter of fact, we have the inequality

|b_{0} + b_{1}f_{1}(x) + … + b_{n}f_{n}(x)| ≫_{ϵ} \frac{1}{h^{n+ϵ}}

with b_{i} ∈ ℤ and

h := max_{0≤i≤n} |b_{i}|.

The inequality above holds for every ϵ > 0 for a wide class of functions f_{i} : U → ℝ, for almost all x ∈ U, where U ⊂ ℝ^{m} is an open subset. Here we use this inequality for the monomials.

Remark: Simon Kristensen [KR] has recently shown, that the set of all pairs (ξ, η) ∈ ℝ^{2} which fail to be strongly Diophantine has Hausdorff dimension 1. Obviously, if (ξ, η) is strongly Diophantine, then any n-tuple of real numbers, which consists of a set of monomials in ξ and η, is Diophantine. Moreover, (ξ, η) is strongly Diophantine iff (−\frac{ξ}{η} , \frac{1}{η}) is such. We have the following analogue of lemma 4.7 in [W], which will eventually allow us to exploit the strong Diophantine assumption of (α, β).

Lemma 2.7. If (ξ, η) is strongly Diophantine, then it satisfies the following property: for any fixed natural m, there exists K ∈ ℕ, such that if

z_{j} = a_{j}^{2} + b_{j}^{2}ξ^{2} + 2a_{j}b_{j}ξ + b_{j}^{2}η^{2} ≪ M,

and ϵ_{j} = ±1 for j = 1, …, m, with integral a_{j}, b_{j} and if ∑_{j=1}^{m} ϵ_{j}\sqrt{z_{j}} ≠ 0, then

| ∑_{j=1}^{m} ϵ_{j}\sqrt{z_{j}}| ≫ M^{−K},  (15)

where the constant involved in the ” ≫ ” notation depends only on η and m.

The proof is essentially the same as the one of lemma 4.7 from [W], considering

the product Q of numbers of the form ∑_{j=1}^{m} δ_{j}\sqrt{z_{j}} over all possible signs δ_{j}. Here

we use the Diophantine condition of the real tuple (ξ, η) rather than of a single real number.

Proposition 2.8. Let m ∈ ℕ be given. Suppose that Λ = ⟨1, α+iβ⟩, such that the pair (ξ, η) := (−\frac{α}{β}, \frac{1}{β}) is algebraically independent strongly Diophantine, which satisfy the property of lemma 2.7 for the given m, with K = K_{m}. Then if M = O(T^{\frac{1−δ}{Km}} ) for some δ > 0, and if L → ∞ such that L/\sqrt{M} → 0, the following holds:

\frac{M_{Λ, m}}{σ^{m}} = \begin{cases}_{\frac{m!}{2^{m/2}(\frac{m}{2})!}} + O(\frac{log L}{L}), & m is even \\ O(\frac{log L}{L}), & m is odd \end{cases}

Proof. Expanding out (10), we have

M_{Λ, m} = \frac{2^{m}}{β^{m}π^{m}} ∑_{k⃗_{1},..., k⃗_{m}∈Λ^{∗}\\{0\}} ∏_{j=1}^{m} \frac{sin (\frac{π|k⃗_{j}|}{L})ψ̂(\frac{|k⃗_{j}|}{\sqrt{M}} )}{|k⃗_{j}|^{\frac{3}{2}}}

× ⟨ ∏_{j=1}^{m} sin (2π(t + \frac{1}{2L})|k⃗_{1}| + \frac{π}{4} )⟩

(16)

Now,

\begin{aligned}⟨ ∏_{j=1}^{m} & sin (2π(t + \frac{1}{2L})|k⃗_{1}| + \frac{π}{4} )⟩ \\ & = ∑_{ϵ_{j}=±1} \frac{∏_{j=1}^{m} ϵ_{j}}{2^{m}i^{m}} ω̂( − T ∑_{j=1}^{m} ϵ_{j}|k⃗_{j}|)e^{πi ∑_{j=1}^{m} ϵ_{j}((1/L)|k⃗_{j}|+1/4)} \end{aligned}

We call a term of the summation in (16) with ∑_{j=1}^{m} ϵ_{j}|k⃗_{j}| = 0 diagonal, and

off-diagonal otherwise. Due to lemma 2.7, the contribution of the off-diagonal terms is:

≪ ∑_{\begin{matrix}k⃗_{1},..., k⃗_{m}∈Λ^{∗}\\{0\} \\ |k⃗_{1}|, ..., |k⃗_{m}|≤\sqrt{M} \end{matrix}} (\frac{T}{M^{K_{m}}} )^{−A} ≪ M^{m}T^{−Aδ},

for every A > 0, by the rapid decay of ω̂ and our assumption regarding M. Since m is constant, this allows us to reduce the sum to the diagonal terms. In order to be able to sum over all the diagonal terms we need the following analogue of a well-known theorem due to Besicovitch [BS] about incommensurability of square roots of integers.

Proposition 2.9. Suppose that ξ and η are algebraically independent, and

z_{j} = a_{j}^{2} + 2a_{j}b_{j}ξ + b_{j}^{2}(ξ^{2} + η),  (17)

such that (a_{j}, b_{j}) ∈ ℤ_{+}^{2} are all different primitive vectors, for 1 ≤ j ≤ m. Then \{\sqrt{z_{j}}\}_{j=1}^{m} are linearly independent over ℚ.

The last proposition is an immediate consequence of a theorem proved in the appendix of [BL2].

Definition: We say that a term corresponding to {k⃗_{1}, …, k⃗_{m}} ∈ (Λ^{∗} \

{0})^{m} and {ϵ_{j}} ∈ {±1}^{m} is a principal diagonal term if there is a partition

{1, …, m} = ⨆_{i=1}^{l} S_{i}, such that for each 1 ≤ i ≤ l there exists a primitive n⃗_{i} ∈

Λ^{∗} \ {0}, with non-negative coordinates, that satisfies the following property: for every j ∈ S_{i}, there exist f_{j} ∈ ℤ with |k⃗_{j}| = f_{j}|n⃗_{i}|. Moreover, for each

1 ≤ i ≤ l, ∑_{j∈S_{i}} ϵ_{j}f_{j} = 0.

Obviously, the principal diagonal is contained within the diagonal. However, the meaning of proposition 2.9 is, that in our situation, the converse also is true:

Corollary 2.10. Every \underline{diagonal} term is a \underline{principle diagonal} term whenether ξ and η are algebraically independent.

Computing the contribution of the principal diagonal terms is done literally the same way it was done in [W], and we sketch it here. As in [W], one can show

that the contribution of a particular partition {1, …, m} = ⨆_{i=1}^{l} S_{i} is negligible,

unless m = 2l is even and #S_{i} = 2 for all 1 ≤ i ≤ l. In the latter case, the contribution is asymptotic to 1. Therefore, the mth moment is asymptotic to 0, if m is odd, and to the number of partitions

{1, …, m} = ⨆_{i=1}^{l} S_{i} with #S_{i} = 2 for all i, m = 2l. This number equals to

\frac{m!}{2^{m/2}(\frac{m}{2})!}, which is also the m-th moment of the standard Gaussian distribution.

3 Bounding the number of close pairs of lattice points

Roughly speaking, we say that a pair of lattice points, n and n′ is close, if ||n| − |n′|| is small. We would like to show that this phenomenon is rare. This is closely related to the Oppenheim conjecture, as |n|^{2} − |n′|^{2} is a quadratic form on the coefficients of n and n′. In order to establish a quantative result, we use a technique developed in a paper by Eskin, Margulis and Mozes [EMM]. Note that the proof is unconditional on any Diophantine assumptions.

3.1 Statement of the results

The ultimate goal of this section is to establish the following

Proposition 3.1. Let Λ be a lattice and denote

A(R, δ) := {(k⃗, l⃗) ∈ Λ × Λ : R ≤ |k⃗|^{2} ≤ 2R, |k⃗|^{2} ≤ |l⃗|^{2} ≤ |k⃗|^{2} + δ}. (18)

Then if δ > 1, such that δ = o(R), we have

#A(R, δ) ≪ Rδ · log R

In order to prove this result, we note that evaluating the size of A(R, δ) is equivalent to counting integer points v⃗ ∈ ℝ^{4} with T ≤ ∥v⃗∥ ≤ 2T such that

0 ≤ Q_{1}(v) ≤ δ,

where Q_{1} is a quadratic form of signature (2, 2), given explicitly by

Q_{1}(v⃗) = (v_{1} + v_{2}α)^{2} + (v_{2}β)^{2} − (v_{3} + v_{4}α)^{2} − (v_{4}β)^{2}.  (19)

For a fixed δ > 0 and a large R, this situation was considered extensively by Eskin, Margulis and Mozes [EMM]. The authors give an asymptotical upper bound in this situation. We will examine how the constants involved in their bound depend on δ, and find out that there is a linear dependency, which is what we essentially need. The author wishes to thank Alex Eskin for his assistance with this matter.

Remarks: 1. In a more recent paper, Eskin Margulis and Mozes [EMM1] prove that for ”generic” lattice Λ, there is a constant c > 0, such that for any fixed δ > 0, as R → ∞, #A(R, δ) is asymptotic to cδR.

2. For our purposes we need a weaker result:

#A(R, δ) ≪_{ϵ} Rδ · R^{ϵ},

for every ϵ > 0. If Λ is a rectangular lattice (i.e. α = 0), then this result follows from properties of the divisor function (see e.g. [BL], lemma 3.2). Theorem 2.3 in [EMM] considers a more general setting than proposition 3.1. We state here theorem 2.3 from [EMM] (see theorem 3.2). It follows from theorem 3.3 from [EMM], which will be stated as well (see theorem 3.3). Then we give an outline of the proof of theorem 2.3 of [EMM], and inspect the dependency on δ of the constants involved.

3.2 Theorems 2.3 and 3.3 from [EMM]

Let Δ be a lattice in ℝ^{n}. We say that a subspace L ⊂ ℝ^{n} is Δ-rational, if L ∩ Δ is a lattice in L. We need the following definitions:

Definitions:

α_{i}(Δ) := sup \{\frac{1}{d_{Δ}(L)}| L is a Δ − rational subspace of dimension i\},

where

d_{Δ}(L) := vol(L/(L ∩ Δ)).

Also

α(Δ) := max_{0≤i≤n} α_{i}(Δ).

Since the space of unimodular lattices is canonically isomorphic to SL(n, ℝ)/SL(n, ℤ), the notation α(g) makes sense for g ∈ G := SL(n, ℝ). For a bounded function f : ℝ^{n} → ℝ, with |f| ≤ M, which vanishes outside a ball B(0, R), define f̃: SL(n, ℝ) → ℝ by the following formula:

f̃(g) := ∑_{v∈ℤ^{n}} f(gv).

Lemma 3.1 in [S2] implies that

f̃(g) < cα(g),  (20)

where c = c(f) is an explicit constant constant

c(f) = c_{0}M max(1, R^{n}),

for some constant c_{0} = c_{0}(n), independent on f. In section 3.4 we prove a stronger result, assuming some additional information about the support of f. Let Q_{0} be a quadratic form defined by

Q_{0}(v⃗) = 2v_{1}v_{n} + ∑_{i=2}^{p} v_{i}^{2} − ∑_{i=p+1}^{n−1} v_{i}^{2}.

Since

v_{1}v_{n} = \frac{(v_{1} + v_{n})^{2} − (v_{1} − v_{n})^{2}}{2} ,

Q_{0} is of signature p, q. Obviously, G := SL(n, ℝ) acts on the space of quadratic forms of signature (p, q), and discriminant ±1, O = O(p, q) by:

Q^{g}(v) := Q(gv).

Moreover, by the well known classification of quadratic forms, O is the orbit of Q_{0} under this action. In our case the signature is (p, q) = (2, 2) and n = 4. We fix an element h_{1} ∈ G with Q^{h_{1}} = Q_{1}, where Q_{1} is given by (19). There exists a constant τ > 0, such that for every v ∈ ℝ^{4},

τ^{−1}∥v∥ ≤ ∥h_{1}v∥ ≤ τ∥v∥.  (21)

We may assume, with no loss of generality that τ ≥ 1. Let H := Stab_{Q_{0}}(G). Then the natural mophism H\G → O(p, q) is a homeomorphism. Define a 1-parameter family a_{t} ∈ G by:

a_{t}e_{i} = \begin{cases}e^{−t}e_{1}, & i = 1 \\ e_{i}, & i = 2, …, n − 1 \\ e^{t}e_{n}, & i = n \end{cases} .

Clearly, a_{t} ∈ H. Furthermore, let K̂ be the subgroup of G consisting of orthogonal matrices, and denote K := H ∩ K̂. Let (a, b) ∈ ℝ^{2} be given and let Q : ℝ^{n} → ℝ be any quadratic form. The object of our interest is:

V_{(a, b)}(ℤ) = V_{(a, b)}^{Q}(ℤ) = {x ∈ ℤ^{n} : a < Q(x) < b}.

Theorem 2.3 states, in our case:

Theorem 3.2 (Theorem 2.3 from [EMM]). Let Ω = {v ∈ ℝ^{4}| ∥v∥ < ν(v/∥v∥)}, where ν is a nonnegative continuous function on S^{3}. Then we have:

#V_{(a, b)}^{Q_{1}}(ℤ) ∩ T Ω < cT^{2} log T,

where the constant c depends only on (a, b).

The proof of theorem 3.2 relies on theorem 3.3 from [EMM], and we give here a particular case of this theorem

Theorem 3.3 (Theorem 3.3 from [EMM]). For any (fixed) lattice Δ in ℝ^{4},

sup_{t>1} \frac{1}{t} ∫_{K} α(a_{t}kΔ)dm(k) < ∞,

where the upper bound is universal.

3.3 Outline of the proof of theorem 3.2:

Step 1: Define

J_{f}(r, ζ) = \frac{1}{r^{2}} ∫_{ℝ^{2}} f(r, x_{2}, x_{3}, x_{4})dx_{2}dx_{3},  (22)

where

x_{4} = \frac{ζ − x_{2}^{2} + x_{3}^{2}}{2r}

Lemma 3.6 in [EMM] states that J_{f} is approximable by means of an integral over the compact subgroup K. More precisely, there is some constant C > 0, such that for every ϵ > 0,

|C · e^{2t} ∫_{K} f(a_{t}kv)ν(k^{−1}e_{1})dm(k) − J_{f}(∥v∥e^{−t}, Q_{0}(v))ν(\frac{v}{∥v∥})| < ϵ (23)

with e^{t}, ∥v∥ > T_{0} for some T_{0} > 0.

Step 2: Choose a continuous nonnegative function f on ℝ_{+}^{4} = {x_{1} > 0} which vanishes outside a compact set so that

J_{f}(r, ζ) ≥ 1 + ϵ

on [τ^{−1}, 2τ] × [a, b]. We will show later, how one can choose f.

Step 3: Denote T = e^{t}, and suppose that T ≤ ∥v∥ ≤ 2T and a ≤ Q_{0}(h_{1}v) ≤ b. Then by (21), J_{f}(∥h_{1}v∥T^{−1}, Q_{0}(h_{1}v)) ≥ 1+ϵ, and by (23), for a sufficiently large t,

C · T^{2} ∫_{K} f(a_{t}kh_{1}v)dm(k) ≥ 1,  (24)

for T ≤ ∥v∥ ≤ 2T and

a ≤ Q_{0}^{x}(v) ≤ b.  (25)

Step 4: Summing (24) over all v ∈ ℤ^{4} with (25) and T ≤ ∥v∥ ≤ 2T, we obtain:

\begin{aligned}#V_{(a, b)}(ℤ) ∩ [T, 2T]S^{3} & ≤ ∑_{v∈ℤ^{n}} C · T^{2} ∫_{K} f(a_{t}kh_{1}v)dm(k) \\ & = C · T^{2} ∫_{K} f̃(a_{t}kh_{1})dm(k) \end{aligned}  (26)

using the nonnegativity of f.

Step 5: By (20), (26) is

≤ C · c(f) · T^{2} ∫_{K} α(a_{t}kh_{1})dm(k).

Step 6: The result of theorem 2.3 is obtained by using theorem 3.3 on the last expression.

3.4 δ-dependency:

In this section we assume that (a, b) = (0, δ), which suits the definition of the set A(R, δ), (18). One should notice that there only 3 δ-dependent steps:

• Choosing f in step 2, such that J_{f} ≥ 1 + ϵ on [τ^{−1}, 2τ] × [0, δ]. We will construct a family of functions f_{δ} with an universal bound |f_{δ}| ≤ M, such that f_{δ} vanishes outside of a compact set which is only slightly larger than

V (δ) = [τ^{−1}, 2τ] × [−1, −1]^{2} × [0, \frac{δτ}{2} ].  (27)

This is done in section 3.4.1.

• The dependency of T_{0} of step 3, so that the usage of lemma 3.6 in [EMM] is legitimate. For this purpose we will have to examine the proof of this lemma. This is done in section 3.4.2.

• The constant c in (20). We would like to establish a linear dependency on δ. This is straightforward, once we are able to control the number of integral points in a domain defined by (27). This is done in section 3.4.3.

3.4.1 Choosing f_{δ}:

Notation: For a set U ⊂ ℝ^{n}, and ϵ > 0, denote

U_{ϵ} := {x ∈ ℝ^{n} : max_{1≤i≤n} |x_{i} − y_{i}| ≤ ϵ, for some y ∈ U}.

Choose a nonnegative continuous function f_{0}, on ℝ_{+}^{4}, which vanishes outside a compact set, such that its support, E_{f_{0}}, slightly exceeds the set V (1). More precisely, V (1) ⊂ E_{f_{0}} ⊂ V (1)_{δ_{0}} for some δ_{0} > 0. By the uniform continuity of f, there are ϵ_{0}, δ_{0} > 0, such that if max_{1≤i≤4} |x_{i} − x_{i}^{0}| ≤ δ_{0}, then f(x) > ϵ_{0}, for

every x^{0} = (x_{1}^{0}, 0, 0, x_{4}^{0}) ∈ V (1). Thus for (r, ζ) ∈ [τ^{−1}, 2τ] × [0, δ], the contribution of [−δ_{0}, δ_{0}]^{2} to J_{f_{0}} is ≥ ϵ_{0} · (2δ_{0})^{2}. Multiplying f_{0} by a suitable factor, and by the linearity of J_{f_{0}}, we may assume that this contribution is at least 1 + ϵ. Now define f_{δ}(x_{1}, …, x_{4}) := f_{0}(x_{1}, x_{2}, x_{3}, \frac{x_{4}}{δ}). We have for δ ≥ 1

\frac{ζ − x_{2}^{2} + x_{3}^{2}}{2rδ} = \frac{ζ/2r}{δ} − \frac{(x_{2}/\sqrt{δ})^{2}}{2r} + \frac{(x_{3}/\sqrt{δ})^{2}}{2r} .

Thus for δ ≥ 1, if (r, ζ) ∈ [τ^{−1}, 2τ] × [0, δ] and for i = 2, 3, |x_{i}| < δ_{0}, f_{δ} satisfies:

f_{δ}(r, x_{2}, x_{3}, x_{4}) > ϵ_{0},

and therefore the contribution of this domain to J_{f_{δ}} is

≥ ϵ_{0}(2δ)^{2} ≥ 1 + ϵ

by our assumption. By the construction, the family {f_{δ}} has a universal upper bound M which is the one of f_{0}.

3.4.2 How large is T_{0}

The proof of lemma 3.6 from [EMM] works well along the same lines, as long as

f(a_{t}x) ≠ 0  (28)

implies that for t → ∞, x/∥x∥ converges to e_{1} = (1, 0, 0, 0). Now, since a_{t} preserves x_{1}x_{4}, (28) implies for the particular choice of f = f_{δ} in section 3.4.1:

|x_{1}x_{4}| = O(δ); x_{1} ≫ T.

Thus

∥x∥ = x_{1} + O(\frac{δ}{T} ) + O(1),

and so, as long as δ = O(T), x/∥x∥ indeed converges to e_{1}.

3.4.3 Bounding integral points in V_{δ}:

Lemma 3.4. Let V (δ) defined by

V (δ) = [τ^{−1}, 2τ] × [−1, −1]^{n−2} × [0, \frac{δβ}{2} ].  (29)

for some constant τ and n ≥ 3. Let g ∈ SL(n, ℝ) and denote

N(g, δ) := #V (δ) ∩ gℤ^{n}.

Then for δ ≥ 1,

|N(g, δ) − \frac{2^{n−2}(2τ − τ^{−1})δ}{det g} | ≤ c_{5}δ ∑_{i=1}^{n−1} \frac{1}{vol(L_{i}/(gℤ^{n} ∩ L_{i})}

for some g-rational subspaces L_{i} of ℝ^{4} of dimension i, where c_{5} = c_{5}(n) depends only on n.

A direct consequence of lemma 3.4 is the following

Corollary 3.5. Let f : ℝ^{n} → ℝ be a nonnegative function which vanishes outside a compact set E. Suppose that E ⊂ V_{ϵ}(δ) for some ϵ > 0. Then for δ ≥ 1, (20) is satisfied with

c(f) = c_{3} · Mδ,

where the constant c_{3} depends on n only.

In order to prove lemma 3.4, we shall need the following:

Lemma 3.6. Let Λ ⊂ ℝ^{n} be a m-dimensional lattice, and let

A_{t} = (\begin{matrix}1 \\ & 1 \\ & & . & . & . \\ & & & & & t \end{matrix})  (30)

an n-dimensional linear transformation. Then for t > 0 we have

det A_{t}Λ ≤ t det Λ.  (31)

Proof. We may assume that m < n, since if m = n, we obviously have an equality. Let v_{1}, …, v_{m} the basis of Λ and denote for every i, u_{i} ∈ ℝ^{n−1} the vector, which consists of first n − 1 coordinates of v_{i}. Also, let x_{i} ∈ ℝ be the last coordinate of v_{i}. By switching vectors, if necessary, we may assume x_{1} ≠ 0. We consider the function

f(t) := (det A_{t}Λ)^{2},

as a function of t ∈ ℝ. Obviously,

f(t) = det (⟨u_{i}, u_{j}⟩ + x_{i}x_{j}t^{2})_{1≤i, j≤m}.

Substracting \frac{x_{i}}{x_{1}} times the first row from any other, we obtain:

f(t) = | ⟨u_{1}, u_{j}⟩ + x_{1}x_{j}t^{2} ; ⟨u_{2}, u_{j}⟩ − \frac{x_{2}}{x_{1}} ⟨u_{1}, u_{j}⟩ ; ⋮ ; ⟨u_{m}, u_{j}⟩ − \frac{x_{m}}{x_{1}} ⟨u_{1}, u_{j}⟩| ,

and by the multilinearity property of the determinant, f is a linear function of t^{2}. Write

f(t) = a(t^{2} − 1) + bt^{2}.

Thus

b = f(1); a = −f(0),

and so b = det Λ, and a = − det (⟨u_{i}, u_{j}⟩) ≤ 0, being minus the determinant of a Gram matrix. Therefore,

(det A_{t}Λ)^{2} − t^{2} det Λ = a(t^{2} − 1) ≤ 0

for t ≥ 1, implying (30).

Proof of lemma 3.4. We will prove the lemma, assuming β = 2. However, it implies the result of the lemma for any β, affecting only c_{5}. Let δ > 0. Trivially,

N(g, δ) = N(g_{0}, 1),

where g_{0} = A_{δ}^{−1}g with A_{δ} given by (30). Let λ_{1} ≤ λ_{2} ≤ … ≤ λ_{n} be the successive minima of g_{0}, and pick linearly independent lattice points v_{1}, …, v_{n} with ∥v_{i}∥ = λ_{i}. Denote M_{i} the linear space spanned by v_{1}, …, v_{i} and the lattice Λ_{i} = g_{0}ℤ^{n} ∩ M_{i}. First, assume that λ_{n} ≤ \sqrt{τ^{2} + (n − 1)} =: r. Now, by Gauss’ argument,

|N(g_{0}, 1) − \frac{2^{n−1}(2τ − τ^{−1})δ}{det g} | ≤ \frac{1}{det g_{0}} vol(Σ),

where

Σ := {x : dist(x, ∂V (1)) ≤ nλ_{n}}.

Now, for λ_{n} ≤ r,

vol(Σ) ≪ λ_{n},

where the constant implied in the “ ≪ “-notation depends on n only (this is obvious for λ_{n} ≤ \frac{1}{2n}, and trivial otherwise, since for λ_{n} ≤ r, vol(Σ) = O(1)).

Thus,

|N(g_{0}, 1) − \frac{2^{n−1}(2τ − τ^{−1})δ}{det g} | ≪ \frac{λ_{n}}{det g_{0}} ≪ \frac{1}{det Λ_{n−1}}

= \frac{1}{vol(M_{n−1}/M_{n−1} ∩ g_{0}ℤ^{n})} ≤ \frac{δ}{vol(A_{δ}M_{n−1}/A_{δ}M_{n−1} ∩ gℤ^{n})}

Next, suppose that λ_{n} > r. Then,

V (δ) ∩ g_{0}ℤ^{n} ⊂ V (δ) ∩ Λ_{n−1}.

Thus, by the induction hypothesis, the number of such points is:

\begin{matrix}≤ & c_{4} ∑_{i=0}^{k−1} \frac{1}{det(Λ_{i})} = ∑_{i=0}^{k−1} \frac{1}{vol(M_{i}/M_{i} ∩ g_{0}ℤ^{n})} \\ ≤ δ ∑_{i=0}^{k−1} \frac{1}{vol(A_{δ}M_{i}/A_{δ}M_{i} ∩ gℤ^{n})}. \end{matrix}

Since λ_{n} > r, we have

\frac{1}{det g} = \frac{1}{λ_{n}} \frac{1}{det g/λ_{n}} ≪ \frac{1}{det g/λ_{n}} ≪ \frac{1}{λ_{1} · . . . · λ_{n−1}} ,

and we’re done by defining L_{i} := A_{δ}M_{i}.

4 Unsmoothing

4.1 An asymptotic formula for N_{Λ}

We need an asymptotic formula for the sharp counting function N_{Λ}. Unlike the case of the standard lattice, ℤ^{2}, in order to have a good control over the error terms we should use some Diophantine properties of the lattice we are working with. We adapt the following notations: Let Λ = ⟨1, α + iβ⟩, be a lattice, d := det Λ = β its determinant, and t > 0 a real variable. Denote the set of squared norms of Λ by

SN_{Λ} = {|n⃗|^{2} : n ∈ Λ}.

Suppose we have a function δ_{Λ} : SN_{Λ} → ℝ, such that given k⃗ ∈ Λ, there are \underline{no} vectors n⃗ ∈ Λ with 0 < ||n⃗|^{2} − |k⃗|^{2}| < δ_{Λ}(|k⃗|^{2}). That is,

Λ ∩ {n⃗ ∈ Λ : |k⃗|^{2} − δ_{Λ}(|k⃗|^{2}) < |n⃗|^{2} < |k⃗|^{2} + δ_{Λ}(|k⃗|^{2})} = A_{|k⃗|},

where

A_{y} := {n⃗ ∈ Λ : |n⃗| = y}.

Extend δ_{Λ} to ℝ by defining δ_{Λ}(x) := δ_{Λ}(|k⃗|^{2}), where k⃗ ∈ Λ minimizes |x − |k⃗|^{2}| (in the case there is any ambiguity, that is if x = \frac{|n⃗_{1}|^{2}+|n⃗_{2}|^{2}}{2} for vectors n⃗_{1}, n⃗_{2} ∈ Λ with consecutive increasing norms, choose k⃗ := n⃗_{1}). We have the following lemma:

Lemma 4.1. For every a > 0, c > 1,

N_{Λ}(t) = \frac{π}{β} t^{2} − \frac{\sqrt{t}}{βπ} ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|≤\sqrt{N} \end{matrix}} \frac{cos (2πt|k⃗| + \frac{π}{4} )}{|k⃗|^{\frac{3}{2}}} + O(N^{a})

+ O(\frac{t^{2c−1}}{\sqrt{N}} ) + O(\frac{t}{\sqrt{N}} · ( log t + log(δ_{Λ}(t^{2})))

+ O( log N + log(δ_{Λ^{∗}}(t^{2})))

As a typical example of such a function, δ_{Λ}, for Λ = ⟨1, α + iβ⟩, with a Diophantine (α, α^{2}, β^{2}), we may choose δ_{Λ}(y) = \frac{c}{y^{K}}, where c is a constant. In

this example, if Λ ∋ k⃗ = (a, b), then by lemma 2.4, A_{|k⃗|} = ±(a, b), provided that β ∉ ℚ(α).

Sketch of proof. The proof of this lemma is essentially the same as the one of lemma 5.1 in [W]. We start from

Z_{Λ}(s) := \frac{1}{2} ∑_{k⃗∈Λ\0} \frac{1}{|k⃗|^{2s}} = ∑_{(m, n)∈ℤ_{+}^{2}\0} \frac{1}{((m + nα)^{2} + (βn)^{2})^{s}} ,

where the series is convergent for ℜs > 1. The function Z_{Λ} has an analytic continuation to the whole complex plane, except for a single pole at s = 1, defined by the formula

Γ(s)π^{−s}Z_{Λ}(s) = ∫_{1}^{∞}x^{s−1}ψ_{Λ}(x)dx + \frac{1}{d} ∫_{1}^{∞}x^{−s}ψ_{Λ^{∗}}(x)dx − \frac{s − d(s − 1)}{2ds(1 − s)},

where

ψ_{Λ}(x) := \frac{1}{2} ∑_{k⃗∈Λ\0} e^{−π|k⃗|^{2}x}.

Moreover, Z_{Λ} satisfies the following functional equation:

Z_{Λ}(s) = \frac{1}{d}χ(s)Z_{Λ^{∗}}(1 − s),  (32)

with

χ(s) = π^{2s−1} \frac{Γ(1 − s)}{Γ(s)}.  (33)

The connection between N_{Λ} and Z_{Λ} is given in the following formula, which is satisfied for every c > 1:

\frac{1}{2}N_{Λ}(x) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞}Z_{Λ}(s)\frac{x^{s}}{s} ds.

The result of the current lemma follows from moving the contour of the integration to the left, collecting the residue at s = 1 (see [W] for details).

Proposition 4.2. Let a lattice Λ = ⟨1, α + iβ⟩ with a Diophantine triple of numbers (α^{2}, αβ, β^{2}) be given. Suppose that L → ∞ as T → ∞ and choose M, such that L/\sqrt{M} → 0, but M = O(T^{δ}) for every δ > 0 as T → ∞. Suppose furthermore, that M = O(L^{s_{0}}) for some (fixed) s_{0} > 0. Then

⟨|S_{Λ}(t, ρ) − S̃_{Λ, M, L}(t)|^{2}⟩ ≪ \frac{1}{\sqrt{M}}

The proof of proposition 4.2 proceeds along the same lines as the one of proposition 5.1 in [W], using again an asymptotic formula for the sharp counting function, given by lemma 4.1. The only difference is that here we use proposition 3.1 rather than lemma 5.2 from [W]. Once we have proposition 4.2 in our hands, the proof of our main result, namely, theorem 1.1 proceeds along the same lines as the one of theorem 1.1 in [W].

Acknowledgement. This work was supported in part by the EC TMR network Mathematical Aspects of Quantum Chaos, EC-contract no HPRN-CT-2000-00103 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This work was carried out as part of the author’s PHD thesis at Tel Aviv University, under the supervision of prof. Zeév Rudnick. The author wishes to thank Alex Eskin for his help as well as the anonymous referees. A substantial part of this work was done during the author’s visit to the university of Bristol.

References

[BL] Pavel M. Bleher and Joel L. Lebowitz Variance of Number of Lattice Points in Random Narrow Elliptic Strip Ann. Inst. Henri Poincare, Vol 31, n. 1, 1995, pages 27-58.

[BL2] Pavel M. Bleher Distribution of the Error Term in the Weyl Asymptotics for the Laplace Operator on a Two-Dimensional Torus and Related Lattice Problems Duke Mathematical Journal, Vol. 70, No. 3, 1993

[BL3] Bleher, Pavel On the distribution of the number of lattice points inside a family of convex ovals Duke Math. J. 67 (1992), no. 3, pages 461-481.

[BL4] Bleher, Pavel M. ; Lebowitz, Joel L. Energy-level statistics of model quantum systems: universality and scaling in a lattice-point problem J. Statist. Phys. 74 (1994), no. 1-2, pages 167-217.

[BS] A.S. Besicovitch On the linear independence of fractional powers of integers J. London Math. Soc. 15 (1940), pages 3-6.

[EMM] Eskin, Alex; Margulis, Gregory; Mozes, Shahar Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture Ann. of Math. (2) 147 (1998), no. 1, pages 93-141.

[EMM1] Eskin, Alex; Margulis, Gregory; Mozes, Shahar Quadratic forms of signature (2,2) and eigenvalue spaces on rectangular 2-tori , Ann. of Math. 161 (2005), no. 4, pages 679-721

[HB] Heath-Brown, D. R. The distribution and moments of the error term in the Dirichlet divisor problem Acta Arith. 60 (1992), no. 4, pages 389-415.

[HR] C. P. Hughes and Z. Rudnick On the Distribution of Lattice Points in Thin Annuli International Mathematics Research Notices, no. 13, 2004, pages 637-658.

[KM] Kleinbock, D. Y., Margulis, G. A. Flows on homogeneous spaces and Diophantine approximation on manifolds Ann. of Math. (2) 148 (1998), no. 1, pages 339-360.

[KR] S. Kristensen, Strongly Diophantine pairs, private communication.

[MAH] K. Mahler Über das Mass der Menge aller S-Zahlen Math. Ann. 106 (1932), pages 131-139.

[S] Wolfgang M. Schmidt Diophantine approximation Lecture Notes in Mathematics, vol. 785. Springer, Berlin, 1980.

[S2] Schmidt, Wolfgang M. Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height Duke Math. J. 35 1968 327-339.

[W] I. Wigman The distribution of lattice points in elliptic annuli, to be published in Quarterly Journal of Mathematics, Oxford, available online in http://qjmath.oxfordjournals.org/cgi/content/abstract/hai017v1

Igor Wigman

School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel igorv@post.tau.ac.il 24

Documenta Mathematica 11 (2006)