The Journal of Fourier Analysis and Applications

Volume 3, Special Issue, 1997

L a t t i c e P o i n t s

Antonio Cdrdoba

1. Andante: Small Arcs, But Not Too Small A t the beginning there were lat t ice points

and ar i thmet ica l functions l ike the fol lowing: r(n) = #{n = a 2 + b2/a, b ~ Z} = number of latt ice points on the ci rc le x 2 + y2 = n.

I t is well known that

r(n) = O(ne), for every e > 0 .

r(n) ~ O ( ( l o g n ) a ) , for any ce.

On the other hand, let {vi }i=I,2,3 be three latt ice points on the circle o f radius R, centered at

the or igin. The theorem of Hero of A lexandr i a gives us: II Vl - v2 II • II v2 - v3 II • Ilvl - v3 II = 4 R

area ( T ) > 2R. Therefore, length o f the arc Vl~V3 > (2R) 1/3. That is, an arc whose length is (2R) 1/3 contains , at most, two latt ice points. We can state our first result.

Theorem 1. (J. Cilleruelo, A. Cdrdoba) For every or, I / 3 < tx < 1/2, there exists a finite constant c~, so that an arc of length R ~

contains, at most, ca lattice points.

W h a t happens when 1 < ot < 1 ? This is an interest ing open problem. This theorem is of an ar i thmet ical nature. The function r(n) is better unders tood in the r ing of

gauss ian integers and there is an extens ion to Q (4:-L--d), d > 0 square free, where we cons ider now ra (n) = #{n = a 2 + db2/a, b ~ Z}, i.e., lat t ice points on arcs of e l l ipses [2]. In his Ph.D. thesis [3] J i m r n e z has a lso obtained the cor responding vers ion for real quadrat ic fields, that is, lat t ice points on arcs o f hyperbolas .

The fo l lowing is a l ist of c losely related quest ions:

i. Rest r ic t ion lemmas for Four ier Ser ies and integrals.

Math Subject Classifications. Keywords and Phrases. Acknowledgements and Notes. Para Miguel en sus sesenta afios.

1997 CRC Press LLC ISSN I069-5869

860 Antonio Crrdoba

ii. Estimates for Gauss sums.

iii. Existence of infinitely many primes in the sequence {n 2 + 1}.

iv. Problems in the theory of Quantum chaos: Characterization of weak limits of sequences [V:k(x)12dx, where ~k is a normalized eigenfunction of the Laplacian [4].

Sketch of the Proof [1]. Let p be a prime so that p - 1 (4). It can be represented in eight manners as a sum of two

squares p = a 2 + b 2. They correspond to the gaussian integers

a q- bi = ~e2rr i{±~+¼ } t = 0 , 1 , 2 , 3 .

We assign to such a prime p -- 1 (4) an angle q~. An important fact about those angles, corresponding to different primes, is that they are linearly independent over the rationals.

Assume that r(n) > 0 and that

2:k aj n = 2V I-I q k I-I p j

qk-----3(4) pj~l(4)

Lat t ice Points 861

Then r(n) = 4 r - i (1 + otj) and each representation n = a 2 + b 2 yields the gaussian integer

a + bi = ~//ne 2rri{q~O+~yjckj+¼}

where: ~bj is the angle assigned to pj l yjl -< ~j and yj =_ ctj mod (2) t = 0 , 1 ,2 ,3

0 if v is even ~b0= ~ if v is odd

On the circle of radius R = ~ let us assume that an arc of length -¢~R ~ contains m + I lattice points:

We use s = 1 . . . . . m + 1 to label those lattice points and angles:

t $

Given two different points s # s ~, we consider the angle:

S ! ~s,s, = Z Y ; - - Y j t s - - t s'

)

we have:

a) I f t s =- t s' mod (2) then @s.s' is the angle corresponding to a representation of the number

I-Ip ' J

as a sum of two squares.

b) I f t s ~ t s' mod (2) then ~s,s' is the angle corresponding to a representation of

I s _ sr I

2 H p 9 f)

J

The fact that the angles Cj are linearly independent over the rationals allows us to conclude

that ~/s,s' is the angle of a lattice point not on the coordinate axis, but on the circle of radius.

I s s ' I y . - - y . / /

2v/2 [ ' 1 pi - ' - r ' - - where v = 0 or 1. J

Therefore, if III III denotes distance to the integers we have:

1 [ ~s,s' >_ 2zr ~/21-Ij P)

On the other hand our hypothesis about the location of the m + 1 lattice points on an arc of length ~/'2R ~ yields:

111,'"' I1-< °-'

862 Antonio C6rdoba

Multiplying these inequalities all together we get:

R(C~- I) m(~+t) I I 1/4 1 1 - - $ s t lYj--Fj 1'4 ~s;' s s'

l-I,,,, 1-Ij pl 1-1j p~ ~Y; -y;

Now we observe that because of the constrains lyfl ~ ~ej we have:

(m + 1) 2 -- 8 (m) Z lyf - y;' < 2o~j 4 S , S !

where 8(m) = / 0 i f m i s o d d 1 i f m i s e v e n t

Therefore,

i.e.,

..~+n I 1 R(a-U >- a j /2 l-lj Pi

(m+ t) 2 -+(m) 4

(m+ 1)2--$(m) = R 4

which yields,

(1 - tr) m(m + 1) < (ra q- 1) 2 - 6(m)

2 - 4

1 1 iv> - 2 4 [ ~ 1 + 2

The proof for other imaginary quadratic fields Q ( ~ ' - d ) follows a similar strategy. One has to assign angles to ideals, and the non-principal prime ideals complicate the program. But it can be carried out (see [2]). [ ]

2. Allegro: Trigonometric Sums Gaussian sums are an important object in Number Theory. Here we shall consider them in the

particular form S N ( X ) = y ~ e 2zrik:x

N<k<_2N

The following are well-known estimates: i.

} _<k~_<2 e 27tik2x P "~ N 1/2, 1 <_ p < 4 N N

ii. {± }c X : e 2:tik2x >__ N l / 2 c t < - ~

N

for some universal constant C < o¢. . .° 111.

e 27tik2x ".-, N ~ (2~.

Lattice Points 863

An application of the method introduced in the proof of the previous theorem produces:

T h e o r e m 2. (J. Cil leruelo, A . Cdrdoba)

f01 N£a 4 ( ) 1 e 27rin2x dx = 2N 2~ + 0 N 3et-l+E , for every e > 0 if ~ < ot < 1 .

N 1

= 2N 2a+0(1 ) , / f a < ~ .

It is an interesting open problem to extends Theorem 2 to the case of arbitrary coefficients, i.e., sums of the form

N + N ¢~ Z ake2rrik2x

N k Trigonometric series whose frequencies are precisely the powers {n }n=1,2,3,... are relevant in Number

Theory. There is a conjecture about their LP-behavior:

Conjec ture 1. / f ~ l a n l 2 < ~ , then ~-~ane 2rrinkx E LP[0, 1], f o r p < 2k.

In the following we shall consider the family of trigonometric series

Sct.k(X) = ~ Z e 27rinkx • net

n=l

They have an interesting history. According to Weierstrass [5], Riemann thought that ImS2,2(x) could be an example of a continuous but nowhere differentiable function. This was partially con- firmed by Hardy [6], who proved that IMS2,2 is not differentiable at any irrational value of x and at several types of rational values. However, years later Gerver [7] showed that there are infinitely many rational numbers in which the derivative exists.

With the help of a computer we have obtained the graphics in the following pages. These graphics illustrate our next theorems about the fractal (box counting) dimension and

differentiability properties of the functions

Fet,~ ( x ) = C...~n e2:rinkx net n=l

0 < lim inf cn < lira sup cn <

T h e o r e m 3. (F. Chamizo , A . Cdrdoba) I f 1 < ot < k + 1/2, then we have

dim(Fet,k) = 2 + - - -- -- 1 ot

2k k

T h e o r e m 4. (F. Chamizo, A. Cdrdoba)

Let a /q ~ Q be an irreducible fraction and S (a /q ) -~- Zn=lq e2Jri q nk" Then Sk,k(x) is differentiable at a /q i f and only i f S (a /q ) = O. Moreover, in this case,

S~,k(x ) = 2~ri ~-~ ne2~riqnk

q n=l

See [9, 10].

&

IQ

oo

Lattice Points 865

o , . , o , , , , , I o , o o o , ' . o , o o ,

Re F3, 3 : 0 ,5999 s Z s 0,6001 Re F3, 3 : 0,331 < Z < 0 ,336

3. Molto Vivace: The Energy of a Large Atom

As we have seen before, an interesting problem in Number Theory is to estimate trigonometric sums of the following form:

N

k=l

where:

(a) l~ ' (x ) l >_ Co > 0.

(b) /z is a periodic function of average 0.

(c) The amplitude f is usually nice.

E x a m p l e 1.

f - 1, l z ( x ) = e 2~rix, ~b(x) = x 2, then we obtain Gauss sums mod(N).

E x a m p l e 2.

f -- 1, /z(x) = x - [x] - ½, then S represents the error term in the lattice point problem for a curve q~ dilated by N.

y = N$(x /N)

y = (1)(X)

866 Antonio Cdrdoba

In collaboration with Fefferman and Sect, the following example has been analyzed:

/z(x) = dist {x, Z} 2 I 12

¢(f2) = ~ VrF(r) -- r2 ]+ •

f ( £22~-1/2dr [2 P(f2) = VrF(r) -- r2/]+ f ( a ) = P(a----') ' N = Z 1/3

VTZF (r) is the Thomas-Fermi potential for an atom with charge Z which satisfies the perfect scaling:

V : F ( r ) = Z4 /3VrF ( Z I / 3 r ) .

It happens that - ¢ ' ( f 2 ) > Co > 0, so we are in conditions to use Van der Corput's method. For the function q/Q(Z) = Z4/3S(ZI/3), we have obtained the following estimates:

Theorem 5. (A. Cdrdoba, C. Fefferman, L. Sect) a) 3C < ~ , 17~a(z)[ _< C Z 3/2. b) lira sup [Z-3/2~pQ ( Z)I ~ O.

Z-.-+oo

The role of the function ~pQ(Z) in atomic physics is as follows: Consider a non-relativistic atom, consisting of a nucleus of charge Z fixed at the origin and N quantized electrons at positions xi ~ R 3.

N Z 1 The hamiltonian of such a system is given by Hz N = ~ i = l ( - A x i - T~i) + ½ ~jCk acting

on functions ~ e 7-/= AN=iL2(R 3 ® Z2) We define the energy

E(Z) = i~vf E(Z; N), E(Z; N) = inf < HZ.N~, ~ >

11¢11=1

Then we have the following asymptotic

Lattice Points 867

E(Z)=CTFZ 7/3 + I Z 2 +CDSZ 5/3 + ~Q(Z)+O(Z 5/3-a) a > 0 8 ' "

The rigorous proof of this formula is due to Lieb-Simon (1 ° Term), Hughes (2 ° Term) and Fefferman-Seco (Dirac-Schwinger term).

It has been known that the next correction is not just a power (fractionary) of Z. The work of Schwinger and Fefferman-Seco suggest that it is precisely 7to(z) the right candidate for the next term of the expansion. Then its "oscillatory nature" became rather interesting as a first step in the project of understanding the periodic table from first principles.

The estimates for ~Q follow from Van der Corput's method; that is, stationary Phase and Poisson's summation formula, which allows us to make the connection with the last topic of this article.

4. Scherzo: X-Rays Crystallography The spatial configuration of a crystal is usually obtained throughout X-ray diffraction data.

The standard interpretation assigns diffraction peaks intensities to absolute values of the Fourier transform of the periodic electron density p.

The phase problem asks for the reconstruction ofp from the knowledge of It31. In such general terms it is not well posed, l A rather interesting question is to analyze which kind of "chemical" or "geometrical" information about p is relevant to ensure the reconstruction.

A plausible model for the electronic density of one-dimensional crystals is given by sums of Dirac's deltas:

N +oo

EbJ E'/,+. j----1 --cx)

where bj ~ Z* are positive integers and 0 _< xj < 1. The phase problem asks to locate the positions {xj } (modulo translations or reflections xj =

1 - x j) knowing the absolute values.

N

IF(m)l = ~ bje 2~rim'x~ j=l

m E Z .

Example 3.

where

Then

(Gaussian molecule) Let p be an odd prime and consider the gaussian sum:

p--I Gp(m) = ~'-~ (p )e2rr im~

n = l

( p ) = Legendresymbol.= { + l , nRp - I , nNp .

p--I n2 Gp(m) = E e2Jrim-F = 1 + 2 Z e2~ri~m

n=0 nRp

{ ~,/~ if ( m , p ) = l [Gp(m)I = p if p/m

1This was first observed by Pauling + Shappel (1930) in reference to the mineral bixbyite. Calder6n and Pepinsky (1952) introduced a method to construct different homometric sets, i.e., E ~ F and I~el = I~FI.

868 Antonio C6rdoba

Inverse problem: Are gaussian sums characterized by this property?

Theorem 6. (F. Chamizo, A. Cdrdoba) Let 0 = xl < x2 < . . . < xo < 1 be real numbers and {bj} positive integers such that

] ~---~ b. e2~rixj.m IF(m)l = [j__/-~l J

satisfies:

Then either

[F(m)l = r /f ( p , m ) = l

IF(m)[ = Z b j if p / m

.,

F(m) = ADp + Be2~m~Gp(m)

o r

F ( m ) = A D p ( p ) - F B e 2 ~ r i m ~

for suitables rational numbers A, B, and integer k.

We have used the notation:

Dp(x) = y ~ . ~ - I e2nrinx Dirichletkernel Gp(m) = gaussian sums

Crucial lemma: Let (p = e 2ari/p and consider the field O((p).

L e m m a 1. I f all the algebraic conjugates of w ~ Q((p) have equal modulus, then:

either w =

o r 113

for some rational number B and integer k.

p-I

n=l

Proof . Let a be a generator of the Galois group of the extension [Q((p) : Q]. The hypothesis about the algebraic conjugates yields

or(w) = e2rri~ = ((b)a, (a, b) = 1 113

Taking a*a --= 1 mod (b) we obtain

Two cases:

1.

2.

¢b = (¢g)a* ~ O(¢p)

I f p / b then [Q((p) : Q(~'b)] = ~ yields b = p orb = 2p. 4,(b) '

I f p /~b then Q((p) = Q((p . Cb) = Q((pb) and qb(p) = dp(p . b) ~ b = 1, 2.

Lattice Points 869

i.e., we only have four possibilities

b = l , 2 , p , 2p

and it became easy to check that there exists e, 0 < e < p - 1 so that

to

Let us assume that ~r(~'p) = ~'ff and choose k = e / ( g - 1) in Z~ (g > 1 because (r is a generator of the Galois group).

Then

Similarly

~ k . 113

~(¢,)-k ~r(w) ~-k w

= ~pkg +k(x (w) = e 2zri "

W tO

~ cr(to) = ~ 'p = 4 - 1 .

W

cr2(~pkw) = 4-1

~ ( ~ 7 k • to)

Therefore, o'2(~';kw) = ~ ; k w , i.e.,

¢;-kto M =

where M is the subfield invariant under cr 2. Therefore,

( hER nEN !

where R, N denotes, respectively, the set of quadratic and non-quadratic residues mod(p) . Ifo(~'pkW) = ~pkW then ~ - k w ~ Q.

If a ( ~ - k w ) = - ~ ' ; k w , then we have B = - A and ~';kw is a rational multiple of a gaussian sum. [53

R e f e r e n c e s

[1] Cilleruelo, J. and C6rdoba, A. (1992). Trigonometric polynomials and Lattice points, Proc. Amer. Math. Soc. 115.

[2] Cilleruelo, J. and C6rdoba, A. (1994). Lattice points on ellipses, Duke Math. J., 76(3). [3] Jimtnez, J. (1995). Puntos de coordenadas enteras en hiperbolas, Ph.D. Thesis, U.A.M. [4] Jacobson, D. (1993). Quantum limits on flat tori, Ph.D. Thesis, Princeton University, Prineton, NJ. [5] Weierstrass, K. Ober continuiediche Functionen eines reellen Arguments, die fiir keiren Werth des letzteren einen

bestimmten Differentialquotien besitzen. Mathematische Werke II, 71-74. [6] Hardy, G.H. (1916). Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17. [7] Gerver, J. (1970). The differentiability of the Riemann function in certain rational multiples ofzr. Amer. J. Math.

22.

870 Antonio Cdrdoba

[8] Duistermaat, J. (1991). Selfsimilarity of "Riemann's non-differentiable function". Nieuw. Arch. Wislc (4)9.

[9] Chamizo, E and C6rdoba, A. (1993). The fractal dimension of a family of Riemann's graphs, C.R. Acad. Sci. Paris, Serie J. 317.

[10] Chamizo, E and Crrdoba, A. Differentiability and dimension of some fractal Fourier series, In press, Adv. Math.

[11] Fefferman, C. and Seco, L. (1994). On the Dirac and Schwinger corrections to the ground-state energy of an atom. Adv. Math. 107.

[12] C6rd•ba•A.•Fe•erman• C.•andSec••L.(•995). Wey•sumsandat•micenergy •sci••ati•ns•Rev.Mat.•ber•amer- icana, 11(1).

[13] Calder6n, A. and Pepinsky, R. Computing Methods and the phase problem in X-ray Crystal analysis, Pub. Penns. State College, College Park, P.A.

[14] Rosenblatt, J. (1984). Phase Retrieval, Comra. Math. Phys. 95.

[ 15] Chamizo, F. and C6rdoba, A. Quadratic Residue Molecules, In Journal of Number Theory, vol. 65, N=BA 1, July 1997.

Universidad Aut6nom'a de Madrid