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Diophantine equations and ergodic theorems

Akos Magyar

Abstract

Let (X, ) be a probability measure space and T1, . . . , T n be a fam-ily of commuting, measure preserving invertible transformations onX. Let Q(m1, . . . , mn) be a homogeneous, positive polynomial withinteger coefficients, and consider the averages:

Af(x) =1

rQ() Q(m)=

f(Tm11 , . . . , T

mnn x)

where rQ() denotes the number of integer solutions m = (m1, . . . , mn)of the diophantine equation Q(m) = .

We prove that under a certain non-degeneracy condition on thepolynomial Q(m) and an ergodic condition on the family of transfor-mations T = (T1, . . . , T n) the pontwise ergodic theorem holds, thatis:

lim

Af(x) =

X

f d

for a.e. x X. This means that the solutions sets of the diophantine

equation Q(m) = become uniformly distributed when mapped to thespace X via the transformations T1, . . . , T n.

The proof uses a variant of the Hardy-Littlewood method of expo-nential sums developed by Birch and Davenport and techniques fromharmonic analysis. A key point is the corresponding maximal theorem,which is a discrete analogue of a maximal theorem on n correspondingto the level surfaces of the polynomial Q(x).

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0. Introduction

A fundamental problem in number theory is to determine asymptoti-cally the number of integer solutions m = (m1, . . . , mn) of a diophantineequation Q(m1, . . . , mn) = as through the integers, and Q(m) isa positive polynomial with integer coefficients. A general result of this typefollows from a variant of the Hardy-Littlewood method of exponential sumsdeveloped by Birch [2] and Davenport [4], which is as follows.

Let Q(m1, . . . , mn) be a positive homogeneous polynomial of degree dwith integral coefficients, and suppose that it satisfies the non-degeneracycondition

(0.1) n dim VQ > (d 1)2d

Here VQ = {z Cn : 1Q(z) = . . . nQ(z) = 0} is the complex singu-

lar variety of the polynomial Q. For simplicity well refer to polynomialssatisfying all the above conditions as non-degenerate forms.

Then the following asymptotic formula holds for the number of integersolutions rQ() = |{m Z

n : Q(m) = }|

(0.2) rQ() = cQnd1

q=1

K(q, 0, ) + O(nd1)

for some > 0. The expression K() =

q=1 K(q, 0, ) is called the singularseries, the terms are special cases of (l = 0) the exponential sums

(0.3) K(q,l ,) = qn

(a,q)=1

sZn/qZ

e2ia(Q(s))+sl

q

that is a goes through the reduced residue classes (mod q) and sj goesthrough all residue classes (mod q) for each j. We remark that K(q, 0, ) isa Kloostermann sum if Q(m) is a quadratic form.

The asymptotic formula (0.2) can be valid just under a condition oftype (0.1). Indeed consider the polynomial Q(m) = (m21 + . . . + m

2n)

d/2

(d > 2 even). Then rQ() = 0 unless = d/2, N, and in that case

rQ() = n/21 = n/d2/d. Hence formula (0.2) is never valid. The reason

is that the complex singular variety: VQ = {z Cn

: z21 + . . . + z

2n = 0}has dimension n 1.

It is meaningful only if the singular series is nonzero. It can be shown,that if Q is a non-degenerate form, then there exists an arithmetic progres-sion N and a constant 0 < AQ such that

(0.4) AQ K() , for every

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well refer to such sets as sets of regular values of the polynomial Q. In-

equality (0.4) is true for all large , just under additional assumptions mod-ulo primes. Indeed consider the polynomial Q(m) = md1 +pQ1(m2, . . . mn).For = p1 + s s being a quadratic non-residue, the equation Q(m) = has no solution, since d is even. Such conditions will be discussed later.

A crucial observation of the paper is, that a similar approximation for-mula to (0.2) holds for the Fourier transform of the solution set:

Q,() =

mZn,Q(m)=

e2im , n

Here n = n/Zn is the flat torus.

Lemma 1 LetQ(m) be a non-degenerate form, then there exists > 0, s.t.

(0.5) Q,() = cQnd1

q=1

K(q,l ,)

lZn(q l)dQ(

1d ( s/q))) +

+E() , and sup

|E()| cnd1

Here () is a smooth cut-off, () = 1 for supj |j| 1/8 and() = 0for supj |j| 1/4 . Moreover

(0.6) dQ() =

{xn :Q(x)=1}

e2ix dQ(x)

here dQ(x) =dSQ(x)|Q(x)| , where dSQ(x) denotes the Euclidean surface area

measure of the level surface Q(x) = 1, and |Q(x)| is the magnitude of thegradient of the form Q.

The approximation formula (0.5) means, that the Fourier transform ofthe indicator function of the solution set Q(m) = is asymptotically a sumover all rational points, of pieces of the Fourier transform of a surface mea-sure of Q(x) = , multiplied by arithmetic factors and shifted by rationals.This formula in the special case Q(m) =

j m

2j was proved earlier in [6].

Our main purpose is to study the distribution of the solution sets

{m Zn

: Q(m) = }.Theorem 1 LetQ(m) be a non-degenerate polynomial and is correspond-ing set of regular values. Then for a test function (x) S(n) one has

(0.7) lim,

1

rQ()

Q(m)=

(1/dm) =

Q(x)=1

(x) dQ(x)

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That is when the solution sets Q(m) = are projected to the unit surface

Q(x) = 1 via the dilations m 1/dm, they weakly converge to the surfacemeasure

dSQ(x)|Q(x)| . This is well-known in case Q(x) is a quadratic form.

The main results of the paper concerns the uniform distribution of theimages of the solution sets, when mapped to a measure space via an ergodicfamily of transformations.

Let (X, ) be a probability measure space, and T = (T1, . . . , T n) be afamily of commuting, measure preserving and invertible transformations.Suppose for every positive integer q the family Tq = (Tq1 , . . . , T

qn) is ergodic.

We recall this means, that for every f L2(X, )

Tq1 f = . . . T qnf = f

implies f = constant. Well refer to a family of transformations satisfyingall the above conditions as a strongly ergodic family.

Theorem 2 Let Q(m) be a non-degenerate form, be a corresponding setof regular values and T = (T1, . . . , T n) a strongly ergodic family of transfor-mations of a measure space (X, ).

For f L2(X, ) consider the averages

Af(x) =1

rQ()

Q(m1,...mn)=

f(Tm11 Tm22 T

mnn x)

Then one has

(0.7) lim,

(Af

Xf d)L2(X,) = 0

This is an L2 ergodic theorem, it follows from a non-trivial estimate onthe exponential sums Q,() at irrational points / Q

n. More preciselyone needs the following

Lemma 2 LetQ(m) be a non-degenerate form, be a corresponding set ofregular values. Then for / Qn one has

(0.8) lim,

1

rQ()|Q,()| = 0

To see the correspondence, suppose that f L2(x, ), f = constant isa joint eigenfunction of the shifts: Tjf = e

2ijf (Tjf(x) = f(Tjx)). ThenAf =

1rQ()

Q,()f , and the strong ergodicity of the family T implies

that / Qn.The main result of the paper is the corresponding pointwise ergodic

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Theorem 3 Let Q(m) be a non-degenerate form, be a corresponding set

of regular values and T = (T1, . . . , T n) a strongly ergodic family of transfor-mations of a measure space (X, ). Let f L2(X, ), Then for -almostevery x X one has

(0.9) lim,

Af(x) =

X

f d

Theorem 3. means, that the images of the solution sets

(0.10) U = {m Zn : Q(m) = }

under the transformations T = (T1

, . . . , T n

) :

(0.11) x, = {(Tm11 T

m22 T

mnn x) : m U}

become uniformly distributed on X w.r.t. for a.e. x X. Let us mentiona special case

Corollary 1 Let 1, . . . , n be a set of irrational numbers (j / Q j).If Q(m) is a non-degenerate form, and is a corresponding set of regularvalues, then the sets

(0.12) , = {(m11, . . . , mnn) n : Q(m1, . . . , mn) = }

become uniformly distributed on the torus n w.r.t. the Lebesgue measure.

Indeed, if X = n and Tj(x1, . . . , xj, . . . xn) (x1, . . . , xj + j , . . . xn)and j / Q, then the family T = (T1, . . . , T n) is strongly ergodic.

The proof of the pointwise ergodic theorem is based on the L2 bound-edness of a corresponding maximal function

Theorem 4 Let Q(m) be a non-degenerate form, be a corresponding setof regular values. For l2(Zn) we define the maximal function

(0.13) N(m) = sup

1rQ()

| Q(l)=

(m l) |

Then one has

(0.14) Nl2(Zn) Cl2(Zn)

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Theorem 4. is a discrete analogue of a maximal theorem on n, corre-

sponding to the level surfaces of the form Q(x).

Theorem 5 LetQ(x) be a non-degenerate form and f L2(n). Then forthe maximal function

(0.15) Mf(x) = sup>0

nd+1|

Q(y)=

f(x y)dSQ,(y)

|Q(y)||

one has

(0.16) MfL2(n) CfL2(n)

For the polynomial Q(x) = nj=1 x2j this is the spherical maximal theo-rem of E.M.Stein [10]. In general, we havent found this result stated in theliterature, nor does it seem to follow easily from the known generalizationsof the spherical maximal theorem, see [8], [9]. In fact the proof will useestimates for exponential sums.

Theorem 4. was proved earlier by Magyar, Stein and Wainger [6], in thespecial case Q(m) =

nj=1 m

2j , moreover there the l

p lp boundedness ofthe discrete maximal operator was shown, for the sharp range of exponents

p > nn2 . The non-degeneracy condition (0.1) is also, sharp in the sense, that

for the form Q(m) = m21 + m22 + m

23 + m

24 (where codim VQ = 4 = (d 1)2

d),

Theorem 4. is not true, taking averages on any arithmetic progression , seesection 5. below. Hence the present work is the continuation of that paperto some extent.

Also we were motivated by Bourgains proof of an ergodic theorem, see[3] corresponding to arithmetic subsets of the natural numbers (such as theset of squares), where the Hardy-Littelwood method was used to reducediscrete maximal operators to the corresponding continuous ones.

However in the present case, the averages are over disjoint sets, thestrong ergodicity condition is also necessary, and is actually a condition onthe joint spectrum of the transformations (T1, . . . , T n). Thus we will needthe Spectral Theorem even in case of the point-wise convergence, i.e. in the

proof of Theorem 3.

1. Exponential sums and oscillatory integrals

We recall some results of Birch [2] on exponential sums, and prove theestimates and properties of oscillatory integrals, needed later. In particularwe give a proof of Theorem 5.

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Let Q(m) be a non-degenerate form of degree d, that is a positive homo-

geneous polynomial with integer coefficients, satisfying the non-degeneracycondition (0.1). Let P > 1, 0 < 1 be fixed.

Definition 1 For 1 q P(d1), 1 a < q, (a, q) = 1 we define themajor arcs

(1.1) La,q() = { : 2| a/q| < q1Pd+(d1)}

L() =

qP(d1), (a,q)=1

La,q()

If / L() then belongs to the minor arcs.

The following properties of the major arcs are immediate from the defi-nition, see [2, Sec.4] for the proof.

Proposition 1 If

(i) 1 < 2 then L(1) L(2)

(ii) < d3(d1) then the intervals La,q() are disjoint for different values

of a and q.

(iii) < d3(d1) then |L()| Pd+3(d1).

Let Q1(m) be a polynomial of degree d, such that its d-degree homoge-neous part Q(m) is a non-degenerate form.

Throughout the paper well use the notation =codim VQ

2d1, and it is

understood that d1 > 2 which follows from condition (0.1). For a real ,and smooth cut-off function (x), consider the exponential sum

(1.2) S() =

mZne2iQ1(m) (m/P)

This is a Weyl type sum, the trivial estimate is S() Pn. The followingestimates due to Birch [2, Sec.4] are of basic importance

Lemma 3 Suppose / L(), then for any > 0, one has

(1.3.1) |S()| CPn+

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If < 2(d1)12d(d1) and2d1 2 < 0. By (1.3)

|S()| CQ,Pn+ = CQ,P

n+(Pd||)

d1 2

The above corollaries can be found in [2, Sec.4-5], however they quicklyfollow from Lemma 3., hence weve included their proofs.

Let Q(x) be a non-degenerate form of degree d, =codim VQ2d1

, L > 0, and n.

Lemma 4 Consider the oscillatory integral

(1.6) IQ(L, ) =

e2i(LQ(x)+x)(x) dx

One has for every > 0

(1.7) IQ(L, ) CQ,(1 + L) d1+

where the constant C is independent of L and .

Proof. The estimate is obvious for L < 1. Let L 1, the gradient ofthe phase: |LQ(x) + | L if || CL on the support of (x) for large

enough constant C > 0, and (1.7) follows by partial integration.Suppose || CL and introduce the parameters P,, s.t. = PdL,

L = P(d1) and P > L3d1 . Changing variables y = P x one has

IQ(L, ) = Pn

e2i (Q(y)+P

d1y)(y/P) dy

We compare the integral to a corresponding exponential sum

PnS() = Pn

mZne2i (Q(m)+P

d1m)(m/P)

If y = m + z where m Zn

and z [0, 1]n

, then

|e2i (Q(y)+Pd1y) e2i (Q(m)+P

d1m)|

C||(|Q(m + z) Q(m)| + Pd1||) CP1+(d1)

since || = Pd+(d1) and || P(d1).

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Thus |IQ(L, ) PnS()| CQP

1+2(d1) CQP 13 . Corollary 3.

implies that

|PnS()| C(Pd)

d1

+CL d1+

and (1.7) follows using P13 L

d1 . 2

Remarks.

i) It is proved in [2, Sec.4] in case = 0, we used a modification of theargument given there.

ii) The proof is based on estimate (1.3), which uses the fact that thepolynomial Q(x) has integer coefficients. Does (1.7) remain true assumingthe coefficients are real ?

iii) In case VQ = {0}, and = 0 the integral decays as (1 + L)nd . What

is the true decay which holds uniformly in , in this case ?

The level surfaces of a non-degenerate form SQ, = {x n : Q(x) = }

are compact smooth hypersurfaces (for > 0). Indeed Q(x) = impliesthat |x| 1/d, moreover Q(x) = 0 for every x = 0.

There is a unique n 1-form dQ(x) on n 0 for which

(1.8) dQ dQ = dx1 . . . dxn

called the Gelfand-Leray form, see [1, Sec.7.1]. To see this, suppose that1Q(x) = 0 on some open set U. By a change of coordinates: y1 =1Q(x), yj = xj for j 2, equation (1.8) takes the form

(1.9) dy1 dQ(y) = 1H(y) dy1 . . . dyn

where x1 = H(y), xj = yj is the inverse map. Thus the form: dQ(y) =1H(y) dy2

. . .

dyn satisfies equation (1.8).

We define the measure dQ, as the restriction of the n 1 form dQto the level surface SQ,. This measure is absolutely continuous w.r.t. the

Euclidean surface are measure dSQ,, more precisely one has

Proposition 2 .

(1.10) dQ,(x) =dSQ,(x)

|Q(x)|

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Proof. Choose local coordinates y as before, in coordinates y level

surface and surface area measure takes the form:

SQ, = {x1 = H(, y2, . . . , yn) : xj = yj}

and

dSQ,(y) = (1 +n

j=2

2j H(, y))1/2 dy2 . . . dyn

Using the identity F(H(y), y2, . . . , yn) = y1 one has

1F(x)1H(y) = 1 , 1F(x)j H(y) + jF(x) = 0

This implies that 1H(y) = (1 + nj=2 2j H(y))1/2 |F(x)|1. Then (1.10)follows by taking y1 = . 2

A key observation of the paper is that the measure dQ,, considered asa distribution on n, has a simple oscillatory integral representation

Lemma 5 Let Q(x) be a non-degenerate form and > 0. Then in thesense of distributions

(1.11) dQ,(x) =

e2i(Q(x))t dt

This means that for any smooth cut-off function(t) and test function(x)

one has

(1.12) lim0

e2i(Q(x))t(t)(x) dxdt =

(x)dQ,(x)

Proof. Let U be an open set on which 1Q = 0, and by a partition ofunity we can suppose, that supp U. Changing variables y1 = Q(x), yj =xj the left side of (1.12) becomes

lim0

e2i(y1)t(t)(y)|1H(y)| dydt =

(, y)|1H(, y

)|dy

where y = (y2, . . . yn).

The last equality can be seen by integrating in t and in y1 first, and usingthe Fourier inversion formula:

lim0

e2i(y1)t(t)g(y1) dy1dt = g()

On the other hand SQ, U = {x1 = H(, y2, . . . yn) : xj = yj } anddQ,(y) = |1H(, y

)| dy in parameters y. 2.

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Lemma 6 Let Q(x) be a non-degenerate form of degree d, =codim VQ2d1

.

Then one has for the Fourier transform of the measure dQ,1 = dQ

(1.13) |dQ()| CQ,(1 + ||) d1+1+

Proof. Suppose || > 1. Using the fact that dQ = dQ if = 1 on aneighborhood of 0 and formula (1.12), we have

(1.14) dQ() =

e2i x(x) dx =

= lim0

e2i xe2i(Q(x)1)t(x)(t) dxdt

We decompose the range of integration into two parts

dQ() =|t|C||

n

+|t|C||

n

= I1 + I2

Since for fixed |t| C|| the gradient of the phase: |tQ(x) | ||/2if C > 0 is small enough, integration by parts gives |I2| CN (1 + ||)

N+1

for every N > 0.For |t| C|| Lemma 3. implies

|

e2i(tQ(x)x)(x) dx| C| |t|

d1

+ hence

I1 C |t|C||

|t|K

d1+ dt C||K

d1+1+

2

First we prove a dyadic version of Theorem 5., together with a refinementwhich will be needed in the proof of Theorem 3.

Lemma 7 Let > 0 be fixed, () be a smooth function with supported on

the set {12d 14}, where = maxj|j|.

Let M and M, be the multipliers acting on L2(n) defined by

Mf() = d(

1/d) and

M,f() = ()d(

1/d

Then one has for the maximal operators

(1.15) sup

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Note that Mf = nd+1 (f d).

Proof. Using the integral representation (1.11) one has

d(1/d) = nd+1[d() =

= nd+1

n

e2i(Q(x))t+m(x/1d ) dxdt

This means

Mf = nd+1

e2itH,tf dt

where H,t is the multiplier corresponding to

h,t() =

e2i(Q(x)t+m)(x/

1d ) dx

Then taking the absolute values, and using Minkowskis integral inequal-ity

(1.16) sup

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Proof of Theorem 5. If Q(x) is a non-degenerate form of degree d,

then the maximal function: M f(x) = sup>0 n/d|Af(x)|, where

Af(x) =

Q(y)

f(x y) dy

is majorized by the standard Hardy-Littlewood maximal function, hence isbounded from L2(n) to itself.

Formula (1.8) means, that for a test function g(y)Q(y)

g(y) dy =0

Q(y)=s

g(y) dQ,s(y)ds

hence

Af(x) = 1

0Af(x) ds

Then the theorem follows by the standard argument of the spherical maximaltheorem, see [10]. 2

2. The approximation formula

First we rewrite formula (0.5) in the form

(2.1) Q,() = cQ

q=1

(a,q)=1

ma/q () + E()

where

(2.2) ma/q () =

lZn

e2ia/qG(a/q,l) (q l)dQ,( l/q)

and G(a/q,l) = qn

sZn/qZn

e2ia(Q(s))+sl

q

Here we used the fact, that dQ,() = n/d1dQ(

1/d), which follows byscaling, since |Q(x)| is homogeneous of degree d 1.

Note that in the right side of (2.1) there is at most one nonzero term,since the cut-off factor (q l), and then (1.4) implies

(2.3) |ma/q ()| C

n/d1q

d1+ C

n/d1q2

by (0.1) if is small enough, hence the sum in (2.1) is absolutely convergent.Let N and M denote the convolution operators on Z

n corresponding

to the multipliers Q,() and m() =

q

(a,q)=1 m

a/q () . The main

approximation property we need is the following

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Lemma 8 Let > 0, > 0 be amall, fixed and f l2(Zn) then

(2.4) sup

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Proof. Let S be defined by Sf = S(, )f(), then

sup

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Next we extend the integration in in (2.9) and define

(2.11) ca/q () =

lZn

G(a,l,q)(q l)I( l/q)

with

(2.12) I( l/q) =

H( l/q,)e2i d

Note that the integral in (2.12) is absolute convergent. Indeed by (1.7)and (0.1)

(2.13) |H(, )| CQ,Pn(1 + Pd||)K

d1+

A crucial point is to identify the the integrals I():

(2.14) I() =

n

e2i(Q(x))e2ix(x/P) d d =

=

n

dQ,(x)e2ix(x/P) d = dQ,()

by (1.11). This means that ca/q () = m

a/q ().

Let A

a/q

, B

a/q

, M

a/q

denote the multipliers, corresponding to a

a/q

(),b

a/q (), and m

a/q ().

Proposition 4 .

(2.15)

qP(d1)

(a,q)=1

sup

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where U is some convolution operator acting on functions on Zn:

Uf =

()f(), and I is some interval. Then one has the point-wise estimate

sup

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but we are integrating now on || Pd+(d1). Note that (q l) = 0 for

at most one values of l, estimate (2.13) and (1.11) : |G(a,l,q)| Cq2.Then

| sup

()| CNPn(1 + Pd||)

Kd1

+

hence by changing variables 1 = Pd one has

sup

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Proof. Note that

Nf() = Q,()f() hence

Nf =q,a

Ma/q f +

q,a

(Aa/q M

a/q )f + E

1f

By Proposition 4. it is enough to show

q,a

sup

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Proposition 6 Let rQ(pN, ) = |{m Zn/pNZn : Q(m) = (mod pN)}|,

that is the number of solutions of the equation Q(m) = (mod pN). Thenone has

(3.2)N

r=0

K(pr, 0, ) = pn(N1)rQ(pN, )

Proof. First

rQ(pN, ) =

m (mod pN)

pNpNb=1

e2i(Q(m)) b

pN

since the inner sum is equal to pN

or 0 according to Q(m) = (mod pN

) ornot. Next one writes b = apNr, where (a, p) = 1, a < pr and r = 0, . . . , N ,and collects the terms corresponding to a fixed r which turn out to beK(pr, 0, ). 2

We remark that this implies: limnpn(N1)rQ(p

N, ) = Kp().To count the number of solutions (mod pN), one uses the p-adic version

of Newtons method, see [7].

Lemma 9 Let p be a prime, and k, l be natural numbers s.t. l > 2k.Suppose there is an m0 Z

n for which

(3.3) Q(m0) (mod pl)

moreover suppose, that pk is the highest power of p which divides all thepartial derivatives jQ(m0).

Then for N l, one has pN(n1)rQ(pN, ) pl(n1)

Proof. For N = l this is obvious. Suppose it is true for N, and con-sider all the solutions m1 (mod p

N+1) of the form m1 = m + pNks where

s (mod p). Then

Q(m +pNks) = Q(m) +pnkQ(m) s = 0 (mod pN+1)

,that is a + b s = 0 (mod p) where apN = Q(m) and bpk = Q(m). Then

bj = 0 (mod p) for some j hence there are pn1 solutions of this form. Allobtained solutions are different mod (pN+1), and m1 satisfy the hypothesisof the lemma. 2

We remark that in case of m = 1, k = 0 the above argument shows thatthere are exactly p(N1)(n1) solutions m for which m = m0 (mod p) andq(m) = (mod pN).

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Lemma 10 let Q(m) be a non-degenerate form, then there exists a set of

regular values in the sense of (0.4).

Proof. Let 0 = Q(m0) = 0 for some fixed m0 = 0. Let p1, . . . , pJ bethe set of primes less then R (R is defined in (3.1)). Let k be an integer s.t.

pkj does not divide d0 , for all j J, where d is degree of Q(m). By the

homogeneity relation Q(m0) m = d0 it follows that pkj does not divide

some partial derivative iQ(m0). Fix l s.t. l > 2k and define the arithmeticprogression

= {0 + kJ

j=1plj : k kQ}. Then we claim that is a set of regular

values. Indeed by Lemma 9. one has for

Kpj

() = limN

pn(N1)

j

rQ(pN

j

, ) pl(N1)

j

This together with (3.1) ensures that the singular series K() remainsbounded from below, and the error term becomes negligible by choosing kQlarge enough. 2

Let us remark that along the same lines it can be shown, that all largenumbers are regular values ofQ(m), if for each prime p < R and each residueclass s (mod p), there is a solution of the equations Q(m) = s (mod p) s.t.Q(m) = 0 (mod p). This is the case for example for Q(m) =

j m

dj .

Let us fix a set of regular values , and a rational point k/p = 0 inn , where k = (k1, . . . , kn) Z

n. Define the measure space X to be the

set of residue classes (mod p), with each element having measure 1/p. LetTj(x) = x + kj (mod p), then the family of transformations T = (T1 . . . T n)is commuting, measure preserving and ergodic. Indeed for some j, kj =0 (mod p) and then Tj is ergodic. The function f(x) = e

2ix/p is a jointeigenfunction : Tjf = e

2ikj/pf hence

(3.5) Af =1

rQ()Q,(k/p)f

where Af are the averages defined in (0.7). Well show below that the meanergodic theorem (0.7) is not valid in this setting, and hence the conditionstrong ergodicity is necessary (note that Tp1 = . . . = T

pn = Id).

Lemma 11 Let be a set of regular values. Let p be a large enough prime:p > d, p > R, p > 0 (where 0 is the smallest element of ), and k Z

n.Then for , = 0 (mod p) one has

(3.6)1

rQ()Q,(k/p) =

1

rQ(p, )

mZn/pZn

e2imkp + O()

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Taking the Lemma granted for a moment, note that the expression:

Sk =

mZn/pZn

e2imkp = 0

for at least one k = 0, since otherwise the equation Q(m) = = 0 (mod p)would have pn or no solution, both cases are impossible (p being largeenough). This follows from Plancherels formula:

k |Sk|

2 = pn|rQ(p, 0)|on the group Zn/pZn. Thus (0.7) is not true, assuming only that the familyof transformations is ergodic.

Proof. For a regular value rQ() = cQK()n/d1+O(n/d1) where

|K()| 1, hence by (0.5), it is enough to show

(3.7) c1Q1

K()

q=1

lZn

K(q,l ,)(qk/p l)dQ(1/d(k/p l/q)) =

=1

rQ(p, )

mZn/pZn

e2imkp + O()

For q not divisible by p, | kp lq |

1pq , hence each term in the sum is

bounded by q

d1+/(d1)+1+ by (1.5) and (1.13). There is at most

one nonzero term in the l sum for fixed q, and thus the total sum for q notdivisible by p is of O().

For q = bp , in (3.7) only those terms for which k/p = l/q are nonzero,hence the sum becomes

1

K()

b=1

K(bp,bk,)

We write q = cpr where (c, p) = 1 and use the multiplicative property

K(cpr+1,ckpr, ) = K(c, 0, )K(pr+1, kpr, )

It is a straightforward computation using the chinese remainder theorem.At this point it is enough to show

(3.8)1

K()(

(c,p)=1

K(c, 0, ))(

r=1

K(pr+1, kpr, )) =

=1

rQ(p, )

m (mod p)

e2imkp

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Again by multiplicativity

(3.9)

(c,p)=1

K(c, 0, )

r=1

K(pr, 0, ) =

q=1

K(q, 0, )

For the other factor in (3.8) one has

(3.10)

r=1

K(pr+1, kpr, ) = p(n1)

m (mod p)

e2imkp

Similarly as in (3.2)

m (mod pN)

pNpNb=1

e2i(Q(m)) b

pN e2imk

pN

and writes b = apNr, where (a, p) = 1, a < pr and r = 0, . . . , N . Eachterm corresponding to a fixed r is K(pr, kpr1, ) for r 1, while the termcorresponding to r = 0 is zero.

Next, let m0 be a solution of Q(m) = (mod p). Then by homogeneityQ(m0) m0 = d = d0 = 0 it follows by the remark after Lemma 9.that the number of solutions: m (mod pN) for which m = m0 (mod p) andQ(m) = (mod pN) is exactly p(n1)(N1). Thus

m (mod pN)

pNpNb=1

e2i(Q(m)) b

pN e2imk

pN = p(n1)

m (modp)

e2imkp

and this proves (3.10).By the same argument

(3.11) Kp() = p(n1)rQ(p, )

and (3.8) follows immediately from (3.9), (3.10) and (3.11). 2

4. TheL

2 ergodic theorem

In this section, we prove Theorems 1-2. and Lemma 2. First we give the

Proof of Theorem 1. Let (x) = (x/1/d), the one has

Q(m)=

(m) =

n

Q,()() d

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where

() d =

mZn(m)e

2im =

mZn(+ m)

by Poisson summation (here () denotes the Fourier transform on n.

Since the exponential sum Q,() is a smooth periodic function on n it

follows

(4.1)

Q(m)=

(m) =

n

Q,()() d

Write Q,() = m() + E() and estimate the contribution of the errorterm

(4.2)

n

|E()()| d Cn/d11 C

n/d1

We used the error estimate in (0.5) and the fact that 1 = 1 C.Recall that

m() =

q=1

l

K(q,l ,)(q l)dQ,( l/q)

Next we estimate the contribution of the terms corresponding to l = 0.For q

12d we use

(4.3)

q12d

l=0

|K(q,l ,)(q l)dQ,( l/q)|

Cn/d1

q12d

q2 Cn/d1

and after integrating we get the same estimate as in (4.2) ( d1 > 2). For

q 12d we give the estimate

(4.4)

q1/2d

l=0

n

|K(q,l ,)(q l)dQ,( l/q) ()| d cNN

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for any N > 0 integer. For fixed l = 0, on the support of the cut-off factor

(q l), one has l/q 1/(4q), which implies 1/(2q), and also l/(2q) (here = supj |j| denotes the sup-norm on

n). Thus

(4.5) |()| CNn/d(1 + 1/d||)2N

Nn/d(1 + 1/d/2q)N(1 + c|l|/2q)N

Integrating in over the region l/q 1/(4q), and then summing in l

and in q 12d one obtains (4.4).

Estimates (4.3) and (4.4) imply together that the total contribution ofthe terms corresponding to l = 0 in (4.1), is O(n/d1.

Finally, we note that

(4.7)

q=1

|K(q, 0, )(1 (q))d()()| d C

nd1

by the same argument as used in proving (4.3) and (4.4). Indeed the rangeof integration is || c/q where both for q 1/2d and for q 1/2d, onehas a gain, using the decay of the the factor K(q, 0, ) for small, and thedecay of for large values of q.

Using (4.3), (4.4) and (4.7) one has

(4.8) n

Q,()() d = cQK() n

Q,()() d+ O(n

d1) =

= rQ()

Q(y)=1

(y) dQ(y) + O(nd1)

Indeed one replaces the singular series cQK() by n/d+1rQ() , use

Plancherels formula, and a change of variables x = 1/dy.This proves the Theorem, since rQ() CQ

n/d1 for regular values .2

Proof of Lemma 2. One writes

(4.9) 1rQ()

|Q,()| Cn/d+1|m()| + O(

)

For q fixed and / Qn (i.e. when j is irrational for some j)

(4.10) n/d+1|mq,()| = cQ

l

|K(q,l ,)(q l)dQ,( l/q)|

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CQq d1+|dQ,(

1/d{q}/q)|

where {} = min | l|. Indeed in the l sum only term corresponding tothe closest lattice point to q is nonzero.

Note that {q} = 0 for every q, since otherwise Qn. Then by(1.13) and (4.10) for q 1/2d we have the estimate n/d+1|mq,()| Cq1, while for q 1/2d one uses the bound q1. The lemmafollows by summing in q. 2

In both the mean and pointwise ergodic theorem the Spectral theoremwill play an essential role. Also, strong ergodidity is a condition on jointspectrum of the shifts Tj (Tjf(x) = f(Tjx)). To see that let (X, ) be

a probability measure space, T = (T1 . . . T n) be a family of commuting,measure preserving and invertible transformations. By the Spectral theoremthere exists a positive Borel measure f on the torus

n, s.t.

(4.11) P(T1, . . . , T n)f, f =n

p()df()

for every polynomial P(z1, . . . , zn), where

p() = p(1, . . . , n) = P(e2i1 , . . . , e2in)

and , denotes the inner product on L2(X, ). We recall two basic facts

i) For r n, f(r) > 0 if and only if r is a joint eigenvalue of the shiftsTj, (i.e. there exists g L

2(X) s.t. Tjg = e2irjg for each j.

ii) If the family T = (T1, . . . , T n) is ergodic, then

f(0) = |f, 1|2 = |

X f d|

2.

Proposition 7 Suppose the family T = (T1, . . . , T n) is ergodic. Then it isstrongly ergodic if and only if f(r) = 0 for every r Q

n, r = 0.

Proof. Suppose f(l/q) > 0 for some l = 0, then there exists g L2(X, ) s.t. Tjg = e

2ilj/qg j. But then Tq

j

g = g j but g = constantsince l = 0.

On the other hand suppose that Tqj g = g, j for some g = constant.Then the functions gs1...sn for s Z

n/qZn defined by

gs1...sn =

mZn/qZn

e2imsq Tm11 . . . T

mnn g

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are joint eigenfunctions of with eigenvalues sj /q. They cannot vanish for all

s = 0 (mod q), because then one would have Tjg = g j, as can be seeneasily by expressing Tjg in terms of the functions gs1...sn. 2

Proof of Theorem 2. We start by

Af f, 1122 = Af

22 |f, 1|

2 =

n/{0}

|Q,()|2

rQ()2df()

The point is that f(Qn/{0}) = 0 by the strong ergodicity condition,

moreover the integrand pointwise tends to zero on the irrationals by Lemma2, and is ma jorized by 1. It follows from the Lebesgue dominant convergencetheorem, that the integral also tends to 0 as . This proves the

theorem.2

5. The discrete spherical maximal theorem

We prove Theorem 4. now. It plays a crucial role in the proof of thepointwise ergodic theorem.

Let l2Zn, the averages we are interested in: 1rQ()

Q(l)= (m l)

will be replaced by

(5.1) N(m) =1

n/d1

Q(l)=

(m l)

Indeed it is enough to prove the maximal theorem for the averages N, since

for regular values: rQ() cQn/d1. We write

(5.2) N = M + E =

q=1

(a,q)=1

Ma/q + E

where M, Ma/q , E denote the mulitpliers corresponding to the functions

n/d+1m(), ma/q (), E(). We denote by M, M

a/q , E the correspond-

ing maximal operators.By Lemma 8.,

(5.2) El2

k=0 sup2k

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Lemma 12 Let q 1, and a s.t. (a, q) = 1 be given. The one has

(5.2) Ma/q l2 Cq

d1+l2

It is understood that Q(m) is a non-degenerate form, hence = 12(d1)

VQ >

2 and > 0 can be taken arbitrary small. Hence in the right side of (5.3)we can take the bound Cq2, but wed like to emphasize the explicitdependence on .

Assuming the Lemma for a moment, by sub-additivity it follows:

Ml2 C

q=1

q q2l2 Cl2

Together with estimate (5.2) this proves Theorem 4.The proof of the lemma is based on a general result, proved in [6]

Lemma 13 Let q 1 be a fixed integer and B be a finite dimensionalBanach space. Let m() be a bounded measurable function on n, takingvalues in B, and supported in the cube [ 12q ,

12q ]

n.Define the periodic extension by

mqper() =

lZnm( l/q)

Let T : L2(n) L2B(n) (where L2B(n) is the space of square in-tegrable functions taking values in the space B), be the multiplier operatorcorresponding to the function m().

Similarly let Tqdis : L2(Zn) L2B(Z

n) be the multiplier operator corre-sponding to the periodic function mqper().

Then one has

(5.4) TqdisL2(Zn)L2B(Zn) CTL2(n)L2B(n)

where the constant C does not depend on the Banach space B, and is alsoindependent of q.

Proof of Lemma 12. Choose a smooth function supported in

[1/2, 1/2]n for which = . Then ma/q () can be written as the product

of the functions

(5.5) ma/q() =

lZnG(a,l,q)( l/q)

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and

(5.6) mq() =

lZn( l/q)d( l/q)

For the first multiplier operator Ma/q it is bounded from l2 to itself withnorm: sup |m

a/q()| Cq d1+.

The sequence of functions mq() defined by (5.6) can be considered asa function mapping from n to the banach space B which is the l

spaceof functions of 1 for some fixed .

The multiplier corresponding to (q)d() is a bounded operator fromL2(n) to L2B(

n) (B being the l space of functions of > 0), which isthe content of Theorem 5. Then one applies Lemma 13. to see that the

multiplier mq() is bounded from l2Zn to l2BZn with norm independent of. This implies (5.2). 2

6. The pointwise ergodic theorem

The proof of Theorem 3. consists of a number of reductions, the argu-ment was motivated by that of Bourgains ergodic theorem corresponding toarithmetic subsets of integers (see [3]). However in our case the averages aretaken over disjoint sets, a condition on the joint spectrum must be imposed,and the Spectral theorem will play an essential role.

Let f L2(X, ), we can suppose

Xf d = 0, and then we have to

show that |Af(x)| 0 for almost every x, as and . Thenagain we can replace the factor rQ() by

n/d1 in the averages.

i) We start with a standard reduction to shifts on Zn. Let (X, ) bea probability measure space, T = (T1, . . . , T n). For x X and L > 0and define: L,x(m) = f(T

mx) if m L and to be 0 otherwise. Herem = (m1, . . . mn) Z

n, m = supj |mj | and Tmx = Tm11 . . . T

mnn x.

Notice that for fixed < L

(6.1) Af(Tlx) = sup

|Af(T

lx)| =

= sup |NL,x(l) = |N

L,x(l)

for l c(L) Thus taking the square, summing in l (for l c(L)),and integrating over the space X one obtains

(6.2) c(L )nAfL2(X)

XN L,xl2 d

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using the fact that the transformations Tl are measure preserving. Also

(6.3)

XL,x

2l2 d = cnL

nf2L2(x)

Then letting , it follows that the L2(X) L2(X) norm of themaximal operator A is majorized by the l

2 l2 norm of the discretemaximal operator N. Then it is enough to prove the pointwise ergodictheorem for a dense subset of L2(X), p.e. for L(X).

ii) Following [5], one reduces pointwise convergence to L2 bounds fortruncated maximal operators. Suppose indirect, that

{x : lim sup |Af(x)| > 0} > 0

then the same is true with a small constant > 0 inserted:

{x : lim sup |Af(x)| > 2} > 2

and using the definition of the upper limit it is easy to see, that to each kif k+1 is chosen large enough then

{x : Akf(x) = supkk+1 |Af(x)| > } >

which implies Akf22 >

3, k . Lets fix such a sequence k which is

quickly increasing: k+1 > 4k4d

. Then it is enough to prove

(6.4)1

K

kK

Akf22 <

3

for K > K(). This means that the Cesaro averages converges in (6.4) tendsto 0 (the terms themselves may not converge to 0).

Now fix K and choose L > K+1. The reasoning in i) leads to

(6.5) c(L )n1

K

kK

Akf2

X

1

K

kK

Nk L,xl2 d

where Nk is defined analogously to Ak. Thus it is enough to prove

(6.6)

X

(1

K

kK

Nk L,x2l2) d cn

3Lnf22

for K > K() and L > L(K, ).

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By (6.3), inequality (6.6) would follow, if the same would be true point-

wise, that is 1/KkK Nk L,x2l2 0 for every x, however this seems tobe true just in average, and has to do with the fact that nearby averagescannot be compared.

i3) We use the approximations to N introduced in Section 2., and thetransfer principle (5.4) to reduce the estimates to that of L2 L2 normsof the corresponding maximal operators acting on n.

We often use the following notations; if() are continuous functions onn, then denote by the corresponding multipliers and by

k the maximal

operator: k=supk

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where per() = l1Zn ( l1) denotes the periodization of . Indeed

write l = ql1 + s and use the fact that G(a,l,q) = G(a,s,q). Again we canfix s (there are at most qn qn choice for each q).

We remark that for l2 and s/q(m) = e2ims/q(m) i.e. s/q() =

(+ s/q), one has

Ms/q,k = Mk s/q

where Ms/q,k is the maximal operator which corresponds to the function

(qs)d(1/d(s/q))per , while Mk corresponds to (q)d(

1/d())per.Indeed one changes variables (s/q) in evaluating the multipliers (the

factors e2/piims/q vanish when taking absolute values).We are in a position to apply the continuous spherical maximal theo-

rem, and further decompose the functions (q)d(1/d()) to get decayestimates. Let

1 = k,0 + k,1 + k,2 be smooth partition of unity on = supj ||j 1/2 such that

k,0() = 0 unless 12

2k+1,

k,1() = 0 unless12

2k+1

12d

k and

k,2() = 0 unless 12dk

Accordingly we have the decomposition: Mk Mk,0 + M

k,1 + M

k,2 and

estimate each term separately.

For fixed , using the fact that |d(1/d) cQ| 1/d|| ( cQ = d(0)),

one has

(6.10) |k,0()(q)d(1/d) cQk,0()(q)| C

1/d2k+1

Thus by the standard square function estimate the l2 l2 norm of themaximal operator (taking the sup over k < k+1) corresponding tothe functions in (6.9) is bounded by:

0) corresponding to the functionsd(1/d()) is bounded from L2 L2 by Theorem 5.

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Thus for s/q = L,x,s/q one has

(6.11) Mk,1s/ql2 CQ

n

|k,1()|2|(+ s/q)|2 d

The point is that since the sequence k is quickly increasing k+1 > 44d

each point can belong to at most 3 intervals Ik on which k,1 supported.Hence averaging over k K the right side of (6.10), gives a contribution of3/K2l2 .

Finally, the family of functions k,2()(q)d(1/d()) satisfy the con-

ditions of Lemma 7. Then (1.16) and (5.4) imply the bound

(6.10) Mk,2s/ql2 CQ 1

2dk l2

Note that (6.9)-(6.11) mean, that the maximal function

1/K

kK

Mk s/q2l2 C

n

|(q)k,1()|2|(+ s/q)|2 d+

+O(K1)s/ql2

i4) It is enough to prove now for fixed r = s/q, that

(6.11) Ln Xn

k,1()|(+ s/q)|2 d d(x)) < ||3f22

ifk > k() and L > L(k, ), where we wrote k() = |k,1()|2 for simplicity

of notation.By applying Plancherel for the inner integral in (6.11), one obtains

Ln

X

m,m

L,x(m) L,x(m)k(m m

)e2i(mm)s/q d d(x) =

= Ln

mL, mL

Tmm

f, fk(m m)e2i(mm

)s/q =

= Lnn

mL, mL

k(m m)e2i(mm

)(+s/q) df() =

= Lnn

lZn

aL(l)k(l)e2i(+s/q) df()

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by the spectral theorem, where aL(l) = |{(m, m); m L, m L, m

m = l}|. Finally one gets

(6.12)

n

(LnaL k) ( + s/q) df

where denotes the convolution on n (w.r.t. Lebesgue measure).Note that

LnaL() = Ln|

Lm=L

e2im|2n Ln min(1,1

L{})2n

This means that L

n

aL is a -sequence (i.e. weakly converges to a Diracdelta) as L . Indeed it is easy to see that: LnaL k ck + forevery > 0 if L is large enough w.r.t. to k and .

Finally if we substitute this estimate into (6.12), then using the fact that

k() = 0 unless 1/2dk , one has

n(LnaL k) ( + s/q) df cdf{ : + s/q <

1/2dk }+

+ df(n) 3f2L2(X)

if k is large enough w.r.t. and L is large enough w.r.t. k and .

Indeed df(n) = f2L2(X), and only here we use the condition strongergodicity, that is the condition that df{s/q} = 0 for every rational points/q = 0 (note that by our assumption df{0} =

X f d = 0 also), which

implies df{ : + s/q < 1/2dk } 0 as k .

This proves Theorem 4. 2.

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References

[1] Arnold, V., Varchenko, A.,: Singularities of differentiable mappingsI-II, Monographs in Math., Birkhauser, Boston (1988)

[2] Birch, B.J.,: Forms in many variable, Proc. Roy. Soc. Ser. A, 265.245-263 (1961)

[3] Bourgain, J.,: On the maximal ergodic theorem for certain subsets ofintegersIsraeli J. Math., 61, 39-72 (1988)

[4] Davenport, H.,: Cubic forms in 32 variables Phil. Trans. A, 251,193-232

[5] Magyar, A.,: Lp

-bounds for spherical maximal operators on Zn

Rev.Mat. Iberoam., 13, 307-317 (1997)

[6] Magyar, A., Stein, E.M., Wainger S.,: Discerete analogues in har-monic analysis: spherical averages, submitted to Annals. of Math.

[7] Serre, J.P.: A course in arithmetic, Graduate texts in Math., SpringerVerlag, (0

[8] Sogge, C. : Fourier integrals in classical analysis, Cambridge Univer-sity Press, (1993)

[9] Sogge, C., Stein, E.M.: Averages of functions over hypersurfaces in

n

, Invent. Math. 82, 543-556, (1985).[10] Stein, E.M., Maximal functions: Spherical means, Proc. Nat. Acad.

Sci., U.S.A., 73, 2174-2175 (1976)

[11] Stein, E.M., Wainger, S: Discerete analogues in harmonic analysisII: fractional integration, Jour. dAnalyse, 80, 335-354 (2000)

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