On the behavior of Lebesgue constants for

two-dimensional Fourier sums over polygons

F.L. Nazarov and A.N. Podkorytov

§0. Introduction.

For every function f ∈ L_{loc}^{1}(ℝ^{2}) which is 1-periodic with respect to each variable one

can define its Fourier series by f(t, s) ∼ ∑_{m,n∈ℤ} f̂(m, n) e^{2πi(mt+ns)} where f̂(m, n) = ∫∫_{[−1/2,1/2]^{2}}

f(t, s) e^{−2πi(mt+ns)} dt ds. Unlike the case of one variable, there is no canonical way to summarize this series. One can choose an arbitrary (closed) bounded domain W ⊂ ℝ^{2} (containing the origin inside it) and put

S_{λ} f(t, s) = ∑_{(m,n)∈λW} f̂(m, n) e^{2πi(mt+ns)}.

This “partial sum” can also be written as a convolution S_{λ} f = f ∗ D_{λ} where D_{λ}(t, s) = ∑_{(m,n)∈λW} e^{2πi(mt+ns)} is the corresponding Dirichlet’s kernel.

The choice of the concrete domain W influences the behavior of the sequence of partial sums S_{λ} f(t, s) very much. We shall restrict ourselves to the case when W is a closed polygon (of very general nature: we assume it neither convex, nor connected; just bounded). This choice is the best possible one in many respects. For example, as it was shown by Fefferman, the Carleson-Hunt theorem on the convergence almost everywhere remains valid (and moreover, is a simple consequence of its one-dimensional version). Our aim is to investigate the growth of the so-called Lebesgue constants L_{λ} =^{def} ∫∫_{[−1/2,1/2]^{2}}

|D_{λ}(t, s)| dt ds when λ → +∞. Let’s recall what has been already known about them

1) for every polygon W c(W) log^{2} λ ≤ L_{λ} ≤ C(W) log^{2} λ for large λ (0 < c(W) < C(W) < +∞)

2) if slopes of all sides of W are rational, D_{λ}(t, s) has almost the same behavior as

its continuous analogue D̃_{λ}(t, s) =∫∫_{λW} e^{2πi(mt+ns)} dm dn; namely, ∫∫_{[−1/2,1/2]^{2}} |D_{λ} −

D̃_{λ}| dt ds ≤ C(W) log λ. The kernel D̃_{λ} is much more handable than D_{λ} in general case and for reasonable domains W the integral of its absolute value over [−1/2, 1/2]^{2} has a good asymptotics not very difficult to calculate. In the case of W being a polygon ∫∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds ∼ c(W) log^{2} λ. The constant c(W) can be easily computed when

W has no parallel sides: it equals \frac{2}{π^{3}} × \{number of sides of W \}. In general case the explicit expression for c(W) is more complicated and we refer the reader to [] for details.

3) In [] the case of W being a triangle W_{α} = {(u, v) : 0 ≤ v ≤ αu ≤ α} was investigated. It was shown that

L_{λ} = ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds + ∫_{0}^{1}|∑_{v=0}^{λ}{αv} e^{2πivt}| dt + O(log λ log log λ)  (∗)

for large λ and that there exists α > 0 for wich the second term is greater than \frac{1}{4π^{2}} log^{2} λ along some (very sparse) sequence of λ tending to +∞ (note that the first term is equivalent to \frac{6}{π^{3}} log^{2} λ). It was shown also that one can choose α in such a way that the ratio \frac{L_{λ}}{log^{2} λ} has no limit when λ → +∞. The main drawback of the results in [] was that they concerned only the values of α belonging to a very thin set and gave no information about other α ∈ (0, +∞) at all. In this paper this uncertainty will be partly overcome: we shall prove that L_{λ} is always essentially greater than ∫∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds unless all the slopes of sides of W are rational, but the question

whether the limit lim_{λ→∞} \frac{L_{λ}}{log^{2} λ} may exist for some polygon W with irrational slopes of sides, remains open. As the authors have different opinions of it, we give no conjecture here.

For reader’s convenience we shall deal with the triangle W_{α} instead of an arbitrary polygon W up to the last paragraph, where we shall describe the changes in proofs to be made and the additional ideas to be involved in general case. It will allow us to avoid clumsy formulas and computations and make the main idea of the proof clear and easy to understand.

§1. Preliminary results.

Here we shall briefly recall the derivation of (∗) in []. The reader who is familiar with it may go to §2 immediately.

We have

\begin{matrix}D_{λ}(t, s) = & ∑_{0≤m≤λ} ∑_{0≤n≤αm} e^{2πi(mt+ns)} = ∑_{0≤m≤λ} e^{2πimt} ∫_{[0,α,n]} e^{2πins} d[n] \\ = & ∑_{0≤m≤λ} e^{2πimt} ∫_{[0,αm]} e^{2πins} dn − ∑_{0≤m≤λ} e^{2πimt} ∫_{[0,αn]} e^{2πins} d\{n\} \\ =^{def} & S_{1}(t, s) − S_{2}(t, s). \end{matrix}

Note that S_{1}(t, s) = \frac{1}{2πis} ∑_{0≤m≤λ} (e^{2πim(t+αs)} − e^{2πimt}) while

D̃_{λ}(t, s) = \frac{1}{2πis} ∫_{0}^{λ}(e^{2πim(t+αs)} − e^{2πimt}) dm.

Beside it, obviously |S_{1}(t, s)| + |D̃_{λ}(t, s)vert ≤ Cλ^{2} for large λ. Thus

\begin{matrix}∫ ∫_{[−1/2,1/2]^{2}} |S_{1} − D̃_{λ}| dt ds = & ∫_{|s|≤ \frac{1}{λ^{2}}} ∫_{|t|≤1/2} + ∫_{1/λ^{2}≤|s|≤ \frac{1}{4α}} ∫_{|t|≤1/2} + ∫_{|s|≥ \frac{1}{4α}} ∫_{|t|≤1/2} \\ =^{def} & I_{1} + I_{2} + I_{3}, \end{matrix}

where

|I_{1}| ≤ ∫_{|s|≤1/λ^{2}} ∫_{|t|≤1/2} Cλ^{2} dt ds ≤ Const,

|I_{3}| ≤ 4α ∫ ∫_{[−1/2,1/2]^{2}} |s| (|S_{1}(t, s)| + |D̃(t, s)|) dt ds ≤ C log λ

(because we integrate the sum of 4 one-dimensional Dirichlet’s kernels). At last, to estimate I_{2}, note that

| ∑_{0≤m≤λ} e^{2πimξ} − ∫_{0}^{λ} e^{2πimξ} dm| ≤ Const for all ξ ∈ (−3/4, 3/4)

and therefore

|I_{2}| ≤ ∫_{1/λ^{2}≤|s|≤1/2} \frac{Const}{|s|} ≤ C log λ

as well and we get

∫ ∫_{[−1/2,1/2]^{2}} |S_{1}(t, s) − D̃_{λ}(t, s)| dt ds = O(log λ) (λ → +∞).

From other side, integrating by parts, we obtain:

S_{2}(t, s) = ∑_{0≤m≤λ} e^{2πimt} + ∑_{0≤m≤λ}{αm} e^{2πim(t+αs)} − ∑_{0≤m≤λ} e^{2πimt} ∫_{0}^{αm} 2πis {n} e^{2πisn} dn.

The first term is just a one-dimensional (one-sided) Dirichlet’s kernel D_{λ}(t) and its L^{1}- norm doesn’t exceed C log λ.

To estimate the L^{1}-norm of the last term, it is convenient to decompose {n} into its Fourier series: \{n\} = \frac{1}{2}+ ∑_{r=0} \frac{1}{2πir} e^{2πirn}. We get

∫_{0}^{α_{m}}\{n\} e^{2πisn} dn = \frac{1}{4πis} (e^{2πimαs} − 1) + ∑_{r≠0} \frac{1}{(2πir) 2πi(r + s)} (e^{2πimα(r+s)} − 1)

and therefore the last term is equal to

\frac{1}{2}(D_{λ}(t + αs) − D_{λ}(t)) + ∑_{r≠0} \frac{s}{2πir(r + s)} (D_{λ}(t + α(r + s)) − D_{λ}(t)).

It is easy to derive from this representation that the L^{1}-norm to be estimated is less than C log λ as well.

Combining all these results, we obtain

D_{λ}(t, s) = D̃_{λ}(t, s) − ∑_{0≤m≤λ}{αm} e^{2πim(t+αs)} + R(t, s)

with ∫∫_{[−1/2,1/2]^{2}} |R(t, s)| dt ds ≤ C log λ.

To derive (∗) from here one is just to note that D̃(t, s) is in essential concentrated at small values of s while the sum ∑_{0≤m≤λ} {αm} e^{2πim(t+αs)} is obviously “equidistributed” on

[−1/2, 1/2] with respect to s in L^{1}-sense. More precisely,

∫_{\frac{1}{log λ} ≤|s|≤1/2} ds(∫_{|t|≤1/2} dt|D̃(t, s)|) ≤ ∫_{\frac{1}{log λ} ≤|s|≤1/2} \frac{C log λ}{|s|} = O(log λ log log λ)

while

∫_{|s|≤ \frac{1}{log λ}} ds(∫_{|t|≤1/2} dt | ∑_{0≤m≤λ}\{αm\} e^{2πim(t+αs)}|)

= \frac{2}{log λ} ∫_{0}^{1}| ∑_{0≤v≤λ}\{αv\} e^{2πivt}| dt = O(log λ)

(the last inequality is well-known and we shall prove it below by pass).

§2. The formulation of the main proposition.

The most part of the information about the integral I_{λ}(α) =^{def} ∫_{0}^{1}| ∑_{0≤v≤λ} {αv} e^{2πivt}|dt

we possess is contained in the following

Proposition.

1) if α = \frac{p}{q} ∈ ℚ then \overline{lim}_{λ→∞} \frac{I_{λ}(α)}{log λ} ≤ sup_{λ≥2} \frac{I_{λ}(α)}{log λ} ≤ C log q;

2) if α ∉ ℚ then c ≤ \overline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} ≤ C, constants c and C being absolute;

3) \underline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} > 0 iff α is not a Liouvillian number, i.e. iff there exists M > 0 such that

|α − \frac{p}{q}| ≥ \frac{const}{q^{M}} for all p, q ∈ ℕ;

4) if α ∉ ℚ and |α − \frac{p}{q}| ≤ \frac{1}{q^{M}} with some M > 2 for infinitely many rationals \frac{p}{q}, then no limit lim_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} exists;

5) the integral I_{λ}(α) is concentrated at a set of small measure, namely, for each λ ≥ 2 and α ∉ ℚ there exists a set E = E(λ, α) ⊂ (0, 1) such that μ(E) ≤ e^{−\sqrt{log λ}} while

∫_{[0,1]\E}| ∑_{0≤v≤λ} {αv} e^{2πivt}| dt ≤ C log^{3/2} λ.

It follows from (2) & (3) that the “typical” behavior of I_{λ}(α) from the viewpoint of category is that \underline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} = 0 while \overline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} ≥ c > 0 (just because the set of non-Liouvillian numbers is of the first category in (0, +∞)).

It is also not very difficult to show that the typical behavior from the viewpoint of Lebesgue measure exists as well. As

∑_{0≤v≤λ/2}\{2αv\} e^{2πi(2v)t} = \frac{1}{2}( ∑_{0≤v≤λ}\{αv\} e^{2πivt} + ∑_{0≤v≤λ}\{αv\} e^{2πiv(t+ \frac{1}{2} )}),

we get I_{λ/2}({2α}) ≤ I_{λ}(α). If we denote

ω(α) =^{def} \underline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ}  and  Ω(α) =^{def} \overline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ},

we obtain immediately from here that ω({2α}) ≤ ω(α) and Ω({2α}) ≤ Ω(α) which (together with the ergodicity of the transform [0, 1) ∋ α ↦ {2α} ∈ [0, 1)) implies ω(α) = ω, Ω(α) = Ω for almost all α ∈ [0, 1) with some 0 < ω ≤ Ω < +∞. Very interesting unsolved question is whether ω = Ω or not.

At last it is worth mentioning that the statements (2)-(4) have their local versions. We shall prove below that for any interval Δ ⊂ ℝ the difference of the integrals of | ∑_{0≤v≤λ}

{αv} e^{2πivt}| over Δ and Δ + α doesn’t exceed C log λ. This (together with (2) and the well-known fact that the numbers {α}, {2α}, {3α},… are equidistributed on (0, 1) when α ∉ ℚ) implies

∫_{Δ}| ∑_{0≤v≤λ}{αv} e^{2πivt}| dt = μ(Δ) I_{λ}(α) + o(log^{2} λ)

where the explicit upper estimate for o(log^{2} λ) depends on α and μ(Δ) only (not on the situation of the interval Δ on the real line). For D_{λ}(t, s) it means that each point (t, s) ≠ (0, 0) gives approximately the same contribution into L_{λ} and therefore no “localization principle” exists (unlike the case of a rational slope, when it is easy to show that the sequence of partial sums S_{λ} f of the function f supported in ε-neighbourhood of the origin converges to 0 outside several strips of width 2ε and even inside these strips S_{λ}(t, s) can grow only as log λ (not as L_{λ} ∼ log^{2} λ) unless (t, s) ∈ supp f.)

Let’s now go to the proofs. We may obviously suppose that α ∈ (0, 1) and λ ∈ ℕ.

§3. Estimates of I_{λ}(α) from above.

First of all consider the case α = \frac{p}{q} with (p, q) = 1. Define 1-periodic function φ(s) by

φ(s) = \{ \begin{matrix}s & , & 0 ≤ s ≤ \frac{q−1}{q} \\ (q − 1)(1 − s) & , & \frac{q−1}{q} ≤ s ≤ 1. \end{matrix}

The Fourier coefficients of φ are φ̂(0) = \frac{q−1}{2q} and φ̂(k) = \frac{q(1−e^{−2πik/q})}{(2πik)^{2}} when k ∈ ℤ, k ≠ 0. It is easy to derive from here that

\frac{1}{π^{2}|k|} ≤ |φ̂(k)| ≤ \frac{1}{2π|k|} for 0 < |k| ≤ q/2 and |φ̂(k)| ≤ \frac{q}{2π^{2}k^{2}}

for all k ∈ ℤ.

Note that

\begin{matrix}∑_{0≤v≤λ}\{αv\} e^{2πivt} = & ∑_{0≤v≤λ} φ(αv) e^{2πivt} = ∑_{0≤v≤λ}(∑_{k∈ℤ} φ̂(k) e^{2πikαv}) e^{2πivt} \\ = & ∑_{k∈ℤ} φ̂(k) ∑_{0≤v≤λ} e^{2πiv(t+kα)} = ∑_{k∈ℤ} φ̂(k) D_{λ}(t + kα) \end{matrix}

(we denote by D_{λ} one-dimensional Dirichlet’s kernel D_{λ}(t) = ∑_{0≤v≤λ} e^{2πivt} again).

Thus,

I_{λ}(α) ≤ (∑_{k∈ℤ} |φ̂(k)|) ∫_{0}^{1} |D_{λ}(t)| dt ≤ C log q log λ,

proving (1).

For any α ∈ (0, 1), λ ≥ 2 one can, according to the well-known Dirichlet theorem, find some rational number \frac{p}{q} such that q ≤ λ and |α − \frac{p}{q}| ≤ \frac{1}{qλ}. Obviously, we may also suppose that (p, q) = 1.

If γ =^{def} α − \frac{p}{q} is positive we have \{αv\} = \{\frac{p}{q} v\} + γv for all 0 ≤ v ≤ λ and therefore

I_{λ}(α) ≤ I_{λ}(\frac{p}{q}) + ∫_{0}^{1}| ∑_{0≤v≤λ} γv e^{2πivt}| dt ≤ C log q log λ + Cγλ log λ ≤ C log^{2} λ + C log λ

for γλ ≤ 1/q ≤ 1.

If γ is negative, the equality \{αv\} = \{\frac{p}{q} v\} + γv still holds, except the values of v divisible by q at which the left and the right part differ by 1. This gives an additional term ∫_{0}^{1} |D_{λ/q}(qt)| dt which is less than C log λ as well, proving a half of (2).

Note also that the sum ∑_{0≤v≤λ} φ̂(k) D_{λ}(t + k \frac{p}{q}) can be considered as a partial sum of

the Fourier series of the (complex-valued) measure ν =^{def} ∑_{k∈ℤ} φ̂(k) δ_{\{−k \frac{p}{q} \}} and therefore

satisfies the weak-type estimate

μ\{t ∈ (0, 1) : |∑_{k∈ℤ} φ̂(k) D_{λ}(t + \frac{kp}{q})| ≥ y\} ≤ \frac{C∥ν∥}{y} ≤ \frac{C log q}{y} ≤ \frac{C log λ}{y}

for all y > 0.

As this sum approximates the sum ∑_{0≤v≤λ} {αv} e^{2πivt} up to the term of L^{1}-norm C log λ

or less, the same weak type etimates holds for ∑_{0≤v≤λ} {αv} e^{2πivt}. So we may put

E(λ, α) =^{def} \{t ∈ (0, 1) : | ∑_{0≤v≤λ}\{αv\} e^{2πivt}| ≥ C log λ e^{\sqrt{log λ}}\},

proving (5).

Suppose at last that |α − \frac{p}{q}| ≤ \frac{1}{q^{M}} with large q and M. Putting λ = q^{M−1}, we get I_{λ}(α) ≤ C log q log λ + C log λ = C(\frac{1}{M−1} + \frac{1}{log λ}) log^{2} λ which is small in comparison with log^{2} λ. Thus, the “only if” part of (3) is also proved.

As for the remark about the difference between the integrals over Δ and Δ + α, it follows immediately from the representation

∑_{0≤v≤λ}{αv} e^{2πivt} − ∑_{0≤v≤λ}{αv} e^{2πiv(t+α)} = ∑_{k∈ℤ} ψ̂(k) D_{λ}(t+αk)+{α(λ+1)} e^{2πi(λ+1)(t+α)},

where ψ(t) is the 1-periodic continuation of the function s(e^{2πis}−1), if we take into account that the Fourier series of ψ is asolutely convergent.

§4. The estimate of I_{λ}(α) from below.

We shall start with the following simple

Lemma.

Let |α − \frac{P}{Q} | ≤ \frac{1}{Q^{2}} and (P, Q) = 1. Then for each λ ∈ (Q^{4}, e^{\sqrt{Q}}) the inequality I_{λ} ≥ c log Q log λ holds with some absolute constant c > 0.

Proof.

Without loss of generality we may suppose that Q is not very small (at any rate we are going to apply this lemma to Q → +∞).

As it was shown in §3, we may approximate ∑_{0≤v≤λ} \{αv\} e^{2πivt} by ∑_{k∈ℤ} φ̂(k) D_{λ}(t+k \frac{p}{q})

for some rational \frac{p}{q} satisfying |α− \frac{p}{q}| ≤ \frac{1}{qλ} and q ≤ λ with function φ corresponding to this denominator q. In the next paragraph it will be very convenient that in this expression D_{λ}(t + k \frac{p}{q}) stands here instead of (more natural) term D_{λ}(t + kα), but now it isn’t so. Nevertheless, as ∫_{0}^{1} |\frac{d}{dt} D_{λ}(t)| dt ≤ Cλ log λ, we have

∑_{k∈ℤ} |φ̂(k)| ∫_{0}^{1}|D_{λ}(t + k \frac{p}{q}) − D_{λ}(t + kα)|

≤ ∑_{|k|≤q/2} |φ̂(k)| |k| |α − \frac{p}{q}| ∫_{0}^{1}|\frac{d}{dt} D_{λ}(t)| dt + 2 ∑_{|k|>q/2} |φ̂(k)| ∫_{0}^{1} |D_{λ}(t)| dt

≤ ∑_{0<|k|≤q/2} \frac{1}{2π|k|} |k| \frac{1}{qλ} Cλ log λ + 2 ∑_{k>q/2} \frac{q}{2π^{2}k^{2}} C log λ ≤ C log λ,

and therefore the approximation by the sum ∑_{k∈ℤ} φ̂(k) D_{λ}(t + kα) gives an error term of

L^{1}-norm C log λ or less as well.

Fix some k_{0} ∈ (0, ^{3} \sqrt{Q}) and consider the interval Δ = Δ(k_{0}) of length \frac{1}{Q^{2}} centered at

−k_{0}α. Note that for each integer ℓ satisfying |ℓ| ≤ \frac{Q}{2} the inequality |ℓα − \frac{ℓP}{Q} | ≤ \frac{|ℓ|}{Q^{2}} ≤ \frac{1}{2Q}

holds and therefore the distance from ℓα to the nearest integer is not less than \frac{1}{Q} − \frac{1}{2Q} = \frac{1}{2Q}.

This, together with the obvious estimate |D_{λ}(ξ)| = |\frac{e^{2πiλξ}−1}{e^{2πiξ}−1} | ≤ \frac{1}{2|ξ|} for |ξ| ≤ 1/2, implies that |D_{λ}(t + kα)| ≤ \frac{1}{1/Q−1/Q^{2}} ≤ 2Q for all t ∈ Δ unless k belongs to some exceptional set Λ ∋ k_{0} satisfying |k′ − k′′| > \frac{Q}{2} for each two different elements k′, k′′ ∈ Λ. Beside it, we get also from here that the intervals Δ(k), k = 1, 2, …, [ ^{3} \sqrt{Q}], are mutually disjoint (in 1-periodic sense).

We have

∫_{Δ}|∑_{k∈ℤ} φ̂(k) D_{λ}(t + kα)| dt ≥ |φ̂(k_{0})| ∫_{Δ} |D_{λ}(t + k_{0}α)| dt

− ∫_{Δ}| ∑_{k∈Λ\{k_{0}}} φ̂(k) D_{λ}(t + kα)| dt − ∫_{Δ}| ∑_{k∈ℤ\Λ} φ̂(k) D_{λ}(t + kα)| dt =^{def} I_{1} − I_{2} − I_{3}.

As λ ≥ Q^{4},

∫_{Δ}|D_{λ}(t + k_{0}α)| dt = ∫_{|t|≤ \frac{1}{2Q^{2}}} |D_{λ}(t)| dt ≥ c log λ

and therefore I_{1} ≥ \frac{c log λ}{π^{2}k_{0}}.

Then, it follows from the above remark immediately that

I_{3} ≤ μ(Δ) 2Q ∑_{k∈ℤ} |φ̂(k)| ≤ \frac{C log q}{Q} ≤ \frac{C log λ}{Q}.

At last,

I_{2} ≤ ∑_{k∈Λ\{k_{0}}} |φ̂(k)| ∫_{0}^{1} |D_{λ}(t)| dt ≤ C log λ ∑_{k∈Λ\{k_{0}}} |φ̂(k)|.

Let’s note that |k| ≥ |k − k_{0}| − |k_{0}| ≥ \frac{Q}{2} − ^{3} \sqrt{Q} > \frac{Q}{3} for each k ∈ Λ\\{k_{0}\}. Thus for some integer d ≥ \frac{Q}{3} elements of Λ\\{k_{0}\} satisfy |k| ≥ d and |k′ − k′′| ≥ d when k′ ≠ k′′. Recall that |φ̂(k)| has a symmetric and decreasing in both sides (with respect to k) majorant

ϕ(k) = \{ \begin{matrix}1/2 & , & k = 0 \\ \frac{1}{2π|k|} & , & 0 < |k| ≤ \frac{q}{2} \\ \frac{q}{2π^{2}k^{2}} & , & |k| > q/2 \end{matrix}

and thereby

∑_{k∈Λ\\{k_{0}\}} |φ̂(k)| ≤ ∑_{k∈Λ\\{k_{0}\}} ϕ(k) ≤ 2 ∑_{ℓ=1}^{∞} ϕ(ℓd) ≤ 2 \frac{1}{d} ∑_{k=1}^{∞} ϕ(k) ≤ \frac{C log q}{d} ≤ \frac{3C log λ}{Q}.

So, we finally have

I_{2} ≤ \frac{C log^{2} λ}{Q} ≤ \frac{C log λ}{\sqrt{Q}} (for λ ≤ e^{\sqrt{Q}})

and

∫_{Δ}|∑_{k∈ℤ} φ̂(k) D_{λ}(t + kα)| dt ≥ (\frac{c}{k_{0}} − \frac{C}{Q} − \frac{C}{\sqrt{Q}}) log λ ≥ \frac{c}{2k_{0}} log λ

with some absolute constants c and C when Q is not very small (we have used here the estimate k_{0} ≤ ^{3} \sqrt{Q} which provides that the first term is actually much greater than both the terms we subtract).

Now

∫_{0}^{1} … ≥ ∑_{1≤k≤ ^{3} \sqrt{Q}} ∫_{Δ(k)} … ≥ ∑_{1≤k≤ ^{3} \sqrt{Q}} \frac{C}{2k} log λ ≥ c′ log Q log λ

and it remains only to note that subtraction of C log λ from the right side part (which is needed to return to the initial integral I_{λ}(α)) doesn’t influence this estimate for large Q. The lemma is proved.

For every α ∉ ℚ one can find infinitely many rationals \frac{P}{Q} satisfying |α − \frac{P}{Q}| ≤ \frac{1}{Q^{2}} . Putting for each such Q λ =^{def} Q^{4}, we get the lower estimate in (2) immediately.

Suppose now that |α − \frac{P}{Q}| ≥ \frac{const}{Q^{M}} for all rational \frac{P}{Q}. For each λ ≥ 1 one can find

some rational \frac{P}{Q} satisfying Q ≤ ^{4} \sqrt{λ}, (P, Q) = 1 and |α − \frac{P}{Q}| ≤ \frac{1}{Q ^{4} \sqrt{λ}} (≤ \frac{1}{Q^{2}} ). For this Q

we must have \frac{1}{Q ^{4} \sqrt{λ}} ≥ \frac{const}{Q^{M}} and therefore Q ≥ (const λ)^{\frac{1}{4(M−1)}} ≥ λ^{\frac{1}{4M}} ≥ log^{2} λ for large λ. Thus all conditions of lemma are satisfied and we get

I_{λ}(α) ≥ c log Q log λ ≥ \frac{c}{4M} log^{2} λ

for large λ, proving the remaining “if” part of (3). Note that both upper and lower estimates for \underline{lim}_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} are of the kind \frac{const}{M}, i.e. this lower limit is inversely proportional to the “Liouvillian exponent” of the number α in essential.

§5. Proof of (4).

First of all note that (4) is a consequence of (2) and the quantitative version of (3) proved above when M is sufficiently large and that the set of α ∈ (0, 1), which can be approximated by rational numbers \frac{p}{q} with an error term essentially less than 1/q^{2}, has zero measure. Thus, the result we are going to prove here is not more than a curious observation. Nevertheless, why not to prove it if we are able to do that?

Suppose the contrary, i.e. that the limit L(α) = lim_{λ→∞} \frac{I_{λ}(α)}{log^{2} λ} exists. Let q be a (very large) denominator for which |α − \frac{p}{q}| ≤ \frac{1}{q^{M}}. It has been shown in §2 that |I_{λ}(α)−I_{λ}(\frac{p}{q})| ≤ C log λ for all 0 ≤ λ ≤ q^{M−1}. We shall consider the values of λ for which λ + 1 is divisible by q (i.e. λ = ℓq − 1, 1 ≤ ℓ ≤ q^{M−2}). The existence of the limit L(α) implies in particular that I_{ℓq−1}(α) = L(α)(log ℓ + log q)^{2} + o(log^{2} q) when q → +∞ (M is fixed!). And we shall show that

I_{ℓq−1}(\frac{p}{q}) = \frac{4}{π^{3}} log q log ℓ + const(q) + o(log^{2} q),

i.e. is a linear function in log ℓ in essential. As the interval of possible values of log ℓ has length compatible with log q, this gives immediately the contradiction desired.

Write again

∑_{0≤v≤ℓq−1}\{\frac{p}{q} v\} e^{2πivt} = ∑_{k∈ℤ} φ̂(k) D_{ℓq−1}(t + k \frac{p}{q}).

Note that for 0 < |k| ≤ \frac{q}{2} the inequality |φ̂(k) − \frac{1}{2πik} | ≤ \frac{C}{q} holds and therefore we may replace this infinite sum by

∑_{{−q/2<k≤q/2 \atop k≠0}} \frac{1}{2πik} D_{ℓq−1}(t + k \frac{p}{q})

(the L^{1}-norm of the error term may be estimated from above by

(|φ̂(0)| + ∑_{0<|k|≤q/2}|φ̂(k) − \frac{1}{2πik} | + ∑_{|k|≥q/2} |φ̂(k)|) ∫_{0}^{1} |D_{ℓq−1}(t)| dt = O(log q) ).

Note also that D_{ℓq−1}(t) = \frac{e^{2πiℓqt}−1}{e^{2πit}−1} and therefore, denoting F (t) =^{def} \frac{1}{e^{2πit}−1}, we can write the last expression in the form

(e^{2πiℓqt} − 1) ∑_{{−q/2<k≤q/2 \atop k≠0}} \frac{1}{2πik} F (t + k \frac{p}{q}).

Let f(t) be the 1-periodic function defined on an interval

Δ_{j} =^{def} (\frac{j}{q} − \frac{1}{2q} , \frac{j}{q} + \frac{1}{2q} )  by  f(t) = \{ \begin{matrix}\frac{1}{2πi(t−j/q)} & when q | j \\ F (j/q) & otherwise. \end{matrix}

It is easy to check that ∫_{0}^{1} |F (t)−f(t)| ≤ 10, say, and thereby we may replace F (t) by f(t), the L^{1}-norm of the corresponding difference being less than 20 ∑_{0<|k|≤q/2} \frac{1}{2π|k|} = O(log q).

Thus we get

\begin{matrix}I_{ℓq−1}(\frac{p}{q}) = & ∫_{0}^{1}|e^{2πiℓqt} − 1| | ∑_{{−q/2<k≤q/2 \atop k≠0}} \frac{1}{2πik} f(t + k \frac{p}{q})| dt + O(log q) \\ = & ∑_{j=0}^{q−1} ∫_{Δ_{j}} … + O(log q). \end{matrix}

Consider each interval Δ_{j} separately. Note that the function f(t + k \frac{p}{q}) is constant on Δ_{j} unless j + kp ≡ 0(mod q) and that for each j = 0, …, q − 1 there is exactly one solution of this comparison which lies between −q/2 and q/2. Therefore, substituting β = q(t − j/q), we obtain

∫_{Δ_{j}} … = ∫_{−1/2}^{1/2}|e^{2πiℓβ} − 1| |−\frac{1}{4π^{2}k_{j}} \frac{1}{β} + b_{j}| dβ

where b_{j} is some constant depending on q and j only (not on ℓ) (this is true for j ≠ 0, when we have even simpler representation without −\frac{1}{4π^{2}k_{j}} \frac{1}{β} under the absolute value sign).

As |e^{2πiℓβ} − 1| is an even function of β, we have

∫_{Δ_{j}} … ≥ ∫_{−1/2}^{1/2} |b_{j}| |e^{2πiℓβ} − 1| dβ = \frac{4}{π} |b_{j}|

and therefore

∑_{j=0}^{q−1} |b_{j}| ≤ \frac{π}{4} I_{q−1}(\frac{p}{q}) + O(log q) ≤ C log^{2} q

(this can be also checked by a direct computation of b_{j}).

As |x + y| and |x| + |y| may differ at most by 2 min(|x|, |y|), we get

\begin{matrix}∫_{Δ_{j}} … = ∫_{−1/2}^{1/2} \frac{1}{4π^{2}|k_{j}|} \frac{|e^{2πiℓβ} − 1|}{|β|} dβ + & ∫_{−1/2}^{1/2} |b_{j}| |e^{2πiℓβ} − 1| dβ \\ + & O(∫_{−1/2}^{1/2} min(\frac{1}{|k_{j}| |β|} , |b_{j}|)). \end{matrix}

The first term is equal to \frac{1}{|k_{j}|} (\frac{2}{π^{3}} log ℓ + O(1)), the second equals to \frac{4}{π} |b_{j}| and at last

∫_{−1/2}^{1/2} min(\frac{1}{|k_{j}| |β|} , |b_{j}|) ≤ ∫_{−1}^{1} … = 2 \frac{1}{|k_{j}|} (1 + log^{+}(|k_{j}| |b_{j}|)).

Note that k_{j} takes all values between −q/2 and q/2 except 0 when j goes from 1 to q − 1 and therefore

I_{ℓq−1}(\frac{p}{q}) = \frac{2}{π^{3}} ( ∑_{0<|k|≤q/2} \frac{1}{|k|}) log ℓ + \frac{4}{π} ∑_{j=0}^{q−1} |b_{j}| + O(log q)

+ O(∑_{j=1}^{q−1} \frac{1}{|k_{j}|} log^{+}(|k_{j}| |b_{j}|)).

The first term may be also written in the form

(\frac{4}{π^{3}} log q + O(1)) log ℓ = \frac{4}{π^{3}} log q log ℓ + O(log q)

(for log ℓ ≤ (M − 2) log q) and therefore to prove the “linearity” desired it is enough to

show that the sum ∑_{j=1}^{q−1} \frac{1}{|k_{j}|} log^{+}(|k_{j}| |b_{j}|) is essentially less than log^{2} q.

Using the inequality log^{+} x ≤ εx + 2 log 1/ε (to check it consider separately the cases 0 ≤ x ≤ 1/ε^{2} and x ≥ 1/ε^{2}), we obtain that this sum is not greater than

ε ∑_{j=1}^{q−1} |b_{j}| + 2 log 1/ε ∑_{j=1}^{q−1} \frac{1}{|k_{j}|} ≤ C(ε log^{2} q + log 1/ε log q)

for every ε > 0. Putting ε = \frac{1}{log q}, we get the estimate of order log q log log q, completing the proof of (4).

§6. What to do with an arbitrary polygon.

Consider any closed polygon W ⊂ ℝ^{2} and choose some horizontal line y = n_{0} ∈ ℤ which lies below it. For every non-vertical side Γ = {(x, y) ∈ ℝ^{2} : a ≤ x ≤ b , y = αx + ν} of W we are able to construct the corresponding trapezoid T (Γ) =^{def} {(x, y) ∈ ℝ^{2} : a ≤ x ≤ b , n_{0} ≤ y ≤ αx + ν}. It is clear that

χ_{W} = ∑_{Γ} ε(Γ) χ_{T (Γ)} + ∑_{Γ:ε(Γ)=−1} χ_{Γ}

(χ_{E} denotes the characteristic function of the set E; ε(Γ) = ±1 depending on whether W and T (Γ) lie on the same side of Γ or not; and, at last, the idle term ∑_{Γ:ε(Γ)=−1} χ_{Γ} is due

to our having assumed both W and T (Γ) closed).

Thus (in natural notations)

D_{W,λ} = ∑_{Γ} ε(Γ) D_{T (Γ),λ} + ∑_{ε(Γ)=−1} D_{Γ,λ} , D̃_{W,λ} = ∑_{Γ} ε(Γ) D̃_{T (Γ),λ}.

The kernels D_{Γ,λ} are actually one-dimensional Dirichlet’s kernels of order λ or less and their contribution may be neglected. The repetition of the argumentation used in §1 gives

D_{T (Γ),λ} = D̃_{T (Γ),λ} − e^{2πiν_{Γ} λs} ∑_{a_{Γ} λ≤m≤b_{Γ} λ}{α_{Γ} m + λ ν_{Γ}} e^{2πim(t+α_{Γ} s)} + “small terms”.

Summarizing over all Γ, we get

D_{W,λ} = D̃_{W,λ} − ∑_{Γ} ε(Γ) e^{2πiν_{Γ} λs} ∑_{a_{Γ} λ≤m≤b_{Γ} λ}{α_{Γ} m + λ ν_{Γ}} e^{2πim(t+α_{Γ} s)} + “small terms”.

It is easy to check that all results of §2 remain valid for the (more general) sum ∑_{aλ≤v≤bλ} {αv + θ} e^{2πivt}. The only difference is that when we try to replace {αv + θ} by

\{\frac{p}{q}v + θ\} + γv, we get several (not more than 1 + |a| + |b|) Dirichlet’s kernels in the error term instead of one that; and the equality \{\frac{p}{q}v + θ\} = φ\{\frac{p}{q} v + θ) doesn’t hold precisely, but gives a Dirichlet kernel as an error as well.

Therefore we are able again to separate the contribution of D̃_{W,λ} into L_{λ} from that of the sum ∑_{Γ} . . .. The property (5) (together with the simple observation that for every

1-periodic function g and every measurable set E ⊂ (0, 1) the inequality

∫_{−1/2}^{1/2}(∫_{E+α′s} |g(t + α′′s| dt) ds ≤ \frac{1 + |α′ − α′′|}{|α′ − α′′|} μ(E) ∫_{0}^{1} |g(t)| dt

holds) allows also to separate the contributions of sides with different slopes.

Finally we get

L_{λ} = ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{W,λ}| dt ds

+ ∑_{α} ∫ ∫_{[−1/2,1/2]^{2}} | ∑_{Γ:α_{Γ} =α} ε(Γ) e^{2πiν_{Γ} λs} ∑_{a_{Γ} λ≤m≤b_{Γ} λ}{α m + λ ν_{Γ}} e^{2πim(t+αs)}| dt ds
+ o(log^{2} λ).

Note also that the same is true if we replace the integration over [−1/2, 1/2]^{2} by that over Δ_{1} × Δ_{2} where Δ_{1} and Δ_{2} are some intervals (with ∫∫_{Δ_{1}×Δ_{2}} |D̃_{W,λ}| dt ds instead of L_{λ} in

the left hand side, of course).

Consider now one fixed group of parallel sides (of slope α, say). Clearly, we may take t + αs as a new variable (which we shall denote by the same letter t). The only difficulty, when returning to old variables t, s, is that the “equidistribution property”

∫ ∫_{Δ_{1}×Δ_{2}} … = μ(Δ_{1} × Δ_{2}) ∫ ∫_{[−1/2,1/2]^{2}} … + o(log^{2} λ)

in new variables will imply that in old variables for slant parallelograms instead of rectangles. But if this property is true for all parallelograms with some fixed slope of sides, it holds for every set measurable in Gordan sense as well! So, we are to investigate the sum

∑_{Γ:α_{Γ} =α} ε(Γ) e^{2πiν_{Γ} λs} f_{Γ,λ}(t)

where

f_{Γ,λ}(t) = ∑_{a_{Γ} λ≤m≤b_{Γ} λ}{α m + λ ν_{Γ}} e^{2πimt}.

We need the following simple fact from the theory of almost-periodic functions

Fact.

Fix some finite set of real exponents ν_{1}, …, ν_{ℓ}. For every ε > 0 one can find T_{0} =

T_{0}(ε; ν_{1}, …, ν_{ℓ}) > 0 such that for any exponential polynomial g(s) = ∑_{j=1}^{ℓ} c_{j} e^{2πiν_{j}s} and for

any interval Δ ⊂ ℝ of length greater than T_{0} the inequality

|\frac{1}{μ(Δ)} ∫_{Δ} |g(s)| ds − M(|g|)| ≤ ε ∑_{j=1}^{ℓ} |c_{j}|

holds, where

M(|g|) =^{def} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} |g(s)| ds

is the “invariant mean” of the function |g| over ℝ.

Proof. (well-known, of course)

Without loss of generality ∑_{j=1}^{ℓ} |c_{j}| = 1. The existence of the invariant mean and

the analogue of the inequality desired is evident for |g(s)|^{2} , |g(s)|^{4}, … instead of |g(s)| just because they are exponential polynomials in fixed sets of exponents with bounded coefficients. It remains only to note that |x| can be uniformly approximated on [−1, 1] by finite linear combinations of |x|^{2} , |x|^{4}, . . ..

As g is determined by its coefficients, we may also denote M(|g|) by M(c_{1}, …, c_{ℓ}). It follows immediately from the fact above that for every interval Δ_{2} ⊂ ℝ

∫_{Δ_{2}} |∑_{Γ} e^{2πiν_{Γ} λs} f_{Γ,λ}(t)| ds = μ(Δ_{2}) M({f_{Γ,λ}(t)}) + o(∑_{Γ} |f_{Γ,λ}(t)|)

when λ → ∞ (M corresponds to the set {ν_{Γ}}_{Γ:α_{Γ} =α} of exponents; {…} is here the “set sign”, not fractional part, of course). Then

∫ ∫_{Δ_{1}×Δ_{2}} |∑_{Γ} e^{2πiν_{Γ} λs} f_{Γ,λ}(t)| dt ds = μ(Δ_{2}) ∫_{Δ_{1}} M({f_{Γ,λ}(t)}) dt + o(log^{2} λ),

for (as it was mentioned above)

∫_{0}^{1} |f_{Γ,λ}(t)| dt = O(log^{2} λ).

Using again the auxiliary function ψ(s) (1-periodic continuation of s(e^{2πis} − 1) from [0, 1]), we get that

∫_{0}^{1}|f_{Γ,λ}(t + α) − e^{−2πiν_{Γ} λ} f_{Γ,λ}(t)| dt ≤ C log λ

and therefore

∫_{Δ_{1}+α} M({f_{Γ,λ}(t)}) dt = ∫_{Δ_{1}} M({e^{−2πiν_{Γ} λ} f_{Γ,λ}(t)}) dt + O(log λ).

To establish the equidistribution property desired, it remains only to note that for every ξ ∈ ℝ and {c_{Γ}} ⊂ ℂ

M({e^{−2πiν_{Γ} ξ} C_{Γ}}) = M({C_{Γ}})

just because the “invariant mean” is invariant under the shift of variable by ξ.

To prove the good analogue of “Proposition”, it is more convenient to keep both variables s and t, not replacing integral over s by the corresponding invariant mean.

We shall again use the approximation

∑_{k∈ℤ} φ̂(k) e^{2πikν_{Γ} λ} ∑_{a_{Γ} λ≤m≤b_{Γ} λ} e^{2πim(t+kα)}

for f_{Γ,λ}(t) and, thereby, the approximation

∑_{k∈ℤ} φ̂(k) ∑_{Γ} ε(Γ) e^{2πiν_{Γ} λ(s+k)} ∑_{a_{Γ} λ≤m≤b_{Γ} λ} e^{2πim(t+kα)}

for the sum to be investigated. Choose some integer ℓ ∈ (\frac{λ}{log λ} , \frac{2λ}{log λ}), say, and split the interval (−1/2, 1/2) into 2ℓ + 1 subintervals

Δ_{j} =^{def} \{t : |t − \frac{j}{2ℓ + 1}| ≤ \frac{1}{4ℓ + 2}\} , j = −ℓ, …, ℓ.

Define the function

F : (−\frac{1}{2} , \frac{1}{2}) → ℂ by F (t) = \{ \begin{matrix}0 & , & t ∈ Δ_{0} \\ \frac{1}{2πi} \frac{2ℓ+1}{j} & , & t ∈ Δ_{j} & , & j ≠ 0 \end{matrix}

(the concrete expression for F (t) doesn’t matter; it is important only that it approximates the denominator \frac{1}{e^{2πit}−1} of the Dirichlet kernel “in L^{1}-sense” and is constant on intervals essentially longer than 1/λ). Denote by D_{λ}(t, s) the 1-periodic continuation (with respect to t, not to s) of the function

F (t) ∑_{Γ} ε(Γ) e^{2πiν_{Γ} λs} (e^{2πib_{Γ} λt} − e^{2πia_{Γ} λt})

(defined initially on (−1/2, 1/2) × ℝ).

It is easy to check that the above sum may be replaced by ∑_{k∈ℤ} φ̂(k) D_{λ}(t + αk, s + k) with

an error about log λ log log λ (in L^{1}-norm). Note that for every partition interval Δ_{j} and for every k ∈ ℤ we have

∫ ∫_{Δ_{j}×[−1/2,1/2]} |D_{λ}(t, s + k)| dt ds

= |F (\frac{j}{2ℓ + 1})| ∫ ∫_{Δ_{j}×([−1/2,1/2]+k)} |∑_{Γ} ε(Γ) e^{2πiν_{Γ} λs} (e^{2πib_{Γ} λt} − e^{2πia_{Γ} λt})| dt ds

= μ(Δ_{j}) |F (\frac{j}{2ℓ + 1})| (M_{α} + o(1))

= (M_{α} + o(1)) ∫_{Δ_{j}} |F (t)| dt  (λ → +∞)

where M_{α} is the invariant mean of the exponential polynomial (in two variables)

G_{α}(t, s) =^{def} ∑_{Γ:α_{Γ} =α} ε(Γ) e^{2πiν_{Γ} s} (e^{2πib_{Γ} t} − e^{2πia_{Γ} t}),

i.e.

M_{α} = lim_{T_{1},T_{2}→∞} \frac{1}{4T_{1}T_{2}} ∫ ∫_{[−T_{1},T_{1}]×[−T_{2},T_{2}]} |G_{α}(t, s)| dt ds

(the existence of M_{α} and the “uniform convergence” property used can be proved in two dimensions just by a repetition of one-dimensional arguments above).

It follows from here immediately that for each ξ ≥ \frac{1}{\sqrt{λ}}, say,

∫ ∫_{[−ξ,ξ]×[−1/2,1/2]} |D_{λ}(t, s + k)| dt ds = (M_{α} + o(1)) \frac{1}{π} log λξ.

Together with the obvious estimate D_{λ}(t, s + k) ≤ \frac{C}{t} (−1/2 ≤ t ≤ 1/2) this allows to carry out all considerations of §§3-5 and obtain an analogue of the “Proposition” in which all absolute constants are replaced by “absolute multiples of M_{α}” (i.e. we write c M_{α} instead of c).

From other side (see [] for details) it is possible to separate the contributions of sides with different slopes into the asymptotics of ∫∫_{[−1/2,1/2]^{2}} |D̃_{λ}(t, s)| dt ds, that of sides with slope

α being precisely \frac{1}{2π^{2}} M_{α} log^{2} λ.

Thus the interference of the contributions of different parallel sides into the “boundary term” is in essential the same as the interference of their contributions into the “integral term” ∫∫_{[−1/2,1/2]} |D̃_{λ}| dt ds and (after bringing all obtained results together and simple

reformulations) we get the following

Proposition 2.

Let W be an arbitrary polygon with slopes of sides α_{1}, …, α_{d} (for vertical sides we assume α = ∞ ∈ ℚ)

I)

∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds = \frac{1}{2π^{2}} (∑_{j=1}^{d} M_{α_{j}} + o(1)) log^{2} λ (λ → ∞)

L_{λ} = ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds + ∑_{j=1}^{d} M_{α_{j}} I′_{λ}(α_{j}) + o(log^{2} λ)

where

I′_{λ}(α_{j}) =^{def} \frac{1}{M_{α_{j}}} ∫ ∫_{[−1/2,1/2]^{2}} | ∑_{Γ:α_{Γ} =α_{j}} ε(Γ) e^{2πiν_{Γ} λs} ∑_{a_{Γ} λ≤m≤b_{Γ} λ}\{α m + λ ν_{Γ}\} e^{2πimt}| dt ds

satisfy (1)-(4) and two-dimentional analog of (5). III)

\overline{lim}_{λ→∞} \frac{1}{log^{2} λ} (L_{λ}− ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds) > 0

unless all α_{j} ∈ ℚ and

\underline{lim}_{λ→∞} \frac{1}{log^{2} λ} (L_{λ}− ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds) = 0

if and only if α_{1}, …, α_{d} can be very good approximated by rationals simultaneously, namely, if for every M > 0 there exist q ∈ ℕ; p_{1}, …, p_{d} ∈ ℤ such that |α_{j} − \frac{p_{j}}{q} | ≤ \frac{1}{q^{M}} for all j = 1, …, d (and, of course, q ≥ 2).

In particular we obtain that for large λ ≥ λ_{0}(W)

c ∫ ∫_{[−1/2,1/2]^{2}} |D̃_{λ}| dt ds ≤ ∫ ∫_{[−1/2,1/2]^{2}} |D_{λ}| dt ds ≤ C ∫ ∫_{[−1/2,1/2]} |D̃_{λ}| dt ds

with some absolute constants c and C. An interesting unsolved problem is whether the same is true for arbitrary not very bad (convex, say) domains W, not only for polygons.