The Uniform Distribution of Sequences Generated by Iteration of Polynomials

Iteration of Polynomials

Emil Lerner

∗

Faulty of Computational Mathematis and Cybernetis,

Mosow State University

Marh 30, 2015

Abstrat

Consideraolletion

f

ofpolynomials

f

i

(

x

)

,

i

= 1

, . . . , s

,withintegeroeients suh that polynomials

f

i(

x

)

−

f

i(0)

,

i

= 1

, . . . , s

, arelinearly independent. Denote by

D

m

the disrepany for the set of points

_f

1

(

x

) mod

m

m

, . . . ,

f

s

(

x

) mod

m

p

n

for all

x

_{∈ {}

0

,

1

, . . . , m

_}

, where

m

=

p

n

,

n

∈

N

, and

p

is a prime number. We prove that

D

m

→

0

as

n

→ ∞

, and

D

m

< c

1

(log log

m

)

−

c

2

, where

c

1

and

c

2

arepositive onstants that depend only on the olletion of

f

i

. As a orollary, we obtain an analogous result for iterations of any polynomial (with integer oeients) whose

degree exeeds 1. Certain results on the uniform distribution were known earlier

onlyfor some lasses ofpolynomials with

s

6

3

.

1 Introdution

The onstrution of pseudorandom generators (PRG) is one of most important

ryp-tographi problems; they have many various pratial appliations. We assume that a

PRG onsistsof

•

a transitionfuntion

f

dening the state of the PRG by the formula

u

i

+1

=

f

(

u

i)

,

where

u

i

isitsstateatthetimemoment

i

(therefore,thestateatthetimemoment

i

is

denedasan

i

-folditerationof thefuntion

f

ofthe initialstate,i.e.,

u

i

=

f

(

i

)

₍

_u

0

)

);

∗

•

anoutput funtion

F

that denes the outputof the PRG atthe time moment

i

as

afuntion of itsurrent state,i.e.,

z

i

=

F

(

u

i)

;

•

the initialstate

u

0

(inwhat follows weassume that itis hosen randomly).

In this paper we study the ability of ertain funtions

f

, namely, polynomials, to

ensurethedesiredpropertyofthesequeneofinternalstates(inotherwords,theabilityto

playthe roleofthe funtion

f

). Weassumethatalulationsare performedmodulosome

number

m

. Forthesakeofuniformityorreasoningwithvarious

m

,weonsiderthenumber

u

i

/m

; evidently, it belongs to the interval from 0 to 1. In order to demonstrate that

onsequent valuesare ¾independent¿ofpreviousones,westudy theset formedbypoints,

whose oordinates are equal to several suññessive values of

u

i

/m

in a multidimensional

unit hyperube.

Withxed

m

the numberof pointsif nite and not greaterthan

m

, beausethe next

state is uniquely dened by the previous one. Therefore, by tending

m

to innity, one

obtainsthe desiredassertions for this ase. Below, as arule,

m

takesthe form of

p

n

with

some prime

p

and natural

n

.

It is well known that for polynomials of degree 1 the measure of the losure of the

mentioned set equals 0, and with

n

tendingtoinnityall pointsbelong toseveral

hyper-planes [2, P. 117℄ inside the unit hyperube. In [4℄ one proves that the measure of the

losure of the orresponding set equals either 0 or 1 for any ompatible funtion

f

, in

partiular, forpolynomialsof any degree (see Theorem 1).

For pratial appliations, along with the unit measure of the losure (i.e., the fat

thatthe

s

-dimensionalubeisoveredbythesetunderonsideration),itisalsoimportant

that the rate (with

n

tending to innity), at whih the ube is being overed by these

points, should be the same at all regions of the ube. More formally, we say that the

projetion of the funtion

f

(

x

)

is uniformly distributed in the

s

-dimensional ube, if for

eah parallelepiped

J

inside the ube the ratio of the number of pointsin

J

to the total

number of points with

n

→ ∞

equals the ratio of the

s

-dimensional volume of

J

to the

total volume of the ube, i.e., to one. See [1℄,[6℄ for denitions of the uniformity for an

arbitraryset of points.

In papers [7℄, [8℄, [9℄oneprovesthe uniformityofthe orrespondingsets forquadrati

polynomials for the number of iterationsof 2 and 3. Moreover, in the mentioned papers

one obtainsonditions under whih the set of pairsof onseutive outputsof aquadrati

generator almost satises the repeated logarithm law [3℄, namely, the prinipal term of

the asymptotis of the disrepany equals

m

−

1

/

2

. In [5℄ these bounds are improved for

the ase of

m

= 2

k

than 2 with integer oeients and an arbitrary number of iterations with

m

=

p

n

,

where

n

tends to innity, and

p

is an arbitrary prime number. The proof is based on

the following evident property: a suiently long random sequene neessarily ontains

any onrete subsequene; moreover, one an hoose the length of the sequene so large

as to make the probability of the opposite event very small. This fat is used in the

indution step. Assumingthe uniformityof the olletion

(

x, x

2

_{, . . . , x}

s

−

1

₎

,we xertain

subsequenes in the number

x

so as to make the major digits of eah funtion of the

olletion

(

x, x

2

_{, . . . , x}

s

₎

modulo

p

n

easily preditable.

2 Basi notions

In this paper we apply tehniques of the

p

-adi analysis for nding funtions that

an be used for onstruting PRG; see, e.g., [10℄ for the neessary denitions. We use

the denitions of the ring of integer

p

-adi numbers and the

p

-adi norm

k · k

p

and

onsider funtions

f

from

Z

p

to

Z

p

. Reall [4℄ that a funtion from

Z

p

to

Z

p

is said to

be ompatible, if

k

f

(

x

1

)

−

f

(

x

2

)

k

p

6

k

x

1

−

x

2

k

p

for any

x

1

, x

2

∈

Z

p

. In other words,

a funtion is ompatible, if for eah

x

1

, x

2

∈

Z

p

, for whih the minor

k

digits in the

p

-adi notation oinide, the minor

k

digits in the

p

-adi notation of

f

(

x

1

)

and

f

(

x

2

)

also

oinide.

For a ompatible funtion

f

(

x

)

and a natural number

s

>

2

the set of points in the

form

x

p

n

,

f

(

x

) mod

p

n

p

n

,

f

(2)

₍

_x

_{) mod}

_p

n

p

n

, . . . ,

f

(

s

−

1)

₍

_x

_{) mod}

_p

n

p

n

exists for all

n

∈

N

, x

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

(hereinafter the denotation

f

(

i

)

₍

_x

₎

means the

i

th iterationof the funtion

f

). We all this set (note that we onsider the union for all

n

) the

s

-dimensionalprojetion of the ompatible funtion

f

.

Consider a funtion

f

having a omplete yle and the orresponding sequene of

states

u

i

=

f

(

u

i

−

1

)

. The set of points inthe form

u

i

mod

p

n

p

n

,

u

i

+1

mod

p

n

p

n

,

u

i

+2

(

x

) mod

p

n

p

n

, . . . ,

u

i

+

s

−

1

(

x

) mod

p

n

p

n

oinides with the set desribed above, beause by the denition of a omplete yle

u

i

runs overall values of

x

.

Inthispaper,insteadofiterationsofonefuntion

f

,asarule,weonsideranarbitrary

olletionof ompatible funtions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s(

x

)

. Let usgeneralize the denition

Let

s

ompatible funtions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s(

x

)

be given. We onsider the set of

points inthe form

f

1

(

x

) mod

p

n

p

n

,

f

2

(

x

) mod

p

n

p

n

, . . . ,

f

s

(

x

) mod

p

n

p

n

forall

x

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

. Forgiven

f

1

, . . . , f

s

andxed

n

wedenotethemultisetunder

onsideration by

P

f

1

,...,f

s

(

n

)

. We allthe union of suh sets for all

n

the joint projetion of funtions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s

(

x

)

.

Inwhat follows weomit subsriptsindiatingolletionsof funtions,ifthey are lear

fromthe ontext.

In [4℄one proves the followingkey theorem:

Theorem 1 (the 0-1 rule). For any ompatible funtion

f

the measure of the losure of

its two-dimensional projetion equals either 0 or 1.

One an easily generalize the mentioned theorem for the ase of arbitrary

s

and an

arbitraryolletionof ompatible funtions

f

1

, f

2

, . . . , f

s

.

Let usnow give amore formaldenition of the projetion uniformity. Let

J

be some

parallelepipedinthe ube

[0

,

1)

s

. Let

F

n

(

J

)

denotethe ratioofthe numberofpointsthat

belongto

P

f

1

,... f

s

(

n

)

and liein

J

tothetotal numberofpoints

p

n

. Let

V

(

J

)

stand forthe

s

-dimensional volume of

J

.

Denition 1. The joint projetion of a olletion of ompatible funtions

f

1

, . . . , f

s

is

said to have a uniform distribution,if

lim

n

→∞

sup

_J

|

V

(

J

)

−

F

n(

J

)

| →

0

,

where the supremum is alulated overall possible parallelepipeds

J

.

In the ase, when as a olletion

f

1

, . . . , f

s

one hooses the set of iterations of some

ompatible funtion

f

(i.e., the set

x, f

(

x

)

, f

(2)

₍

_x

₎

_{, . . . , f}

(

s

−

1)

₍

_x

₎

), we say that the

s

-dimensional projetion funtion

f

has a uniform distribution.

Evidently, the uniformity of the projetion implies that the measure of the losure

equals 1. In this paperwe study the uniformityof projetions of polynomialfuntions

f

.

Considering the supremum for onrete

n

, we obtain the disrepany

D

p

n

. Instead of estimating this value, it is more onvenient to study the lower bound for the digit

apaity,beginningwithwhihtheboundfortheuniformityoftheonsideredsetofpoints

is guaranteed. In this paper we use various denitions of the uniformity (measurable in

terms of various errors

ε

); as a result, they beome onneted with eah other and form

for studyingthe rate of onvergene to0 inDenition 1.

The uniformity of the projetion of a olletion of ompatible funtions means that

for any positive

ε

there exists

N

f

1

,...,f

s

₍

_ε

₎

suh that for any

n > N

f

1

,...,f

s

₍

_ε

₎

it holds

|

F

n

(

J

)

−

V

(

J

)

|

< ε

.

In what follows, when using the denotation

N

f

1

,f

2

,...,f

s

₍

_{k, ε}

_{) =}

_g

₍

_{k, ε}

₎

, we take into

aount the fat that this funtion

g

satises onditions stated in the previous denition,

but the minimum of the estimate isnot guaranteed.

Let

k

be some xed number,

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

₋

₁

_}

. Denote by

J

k

(

a

1

, a

2

, . . . , a

s

)

the hyperubein

[0

,

1)

s

dened by inequalities

a

1

p

k

6

z

1

<

a

1

+ 1

p

k

a

2

p

k

6

z

2

<

a

2

+ 1

p

k

. . .

a

s

p

k

6

z

s

<

a

s

+ 1

p

k

,

where

(

z

1

, z

2

, . . . , z

s)

are oordinates of a point from

[0

,

1)

s

.

Let us slightly modify Denition 1 (f. [1℄; the equivalene of denitions 1 and 2 is

proved inTheorem 2). Namely,

Denition 2. The joint projetion is alled uniformly distributed, if for any number

ε >

0

and for any natural

k

there exists natural

N

f

1

,...,f

s

0

(

k, ε

)

suh that for eah

n >

N

f

1

,...,f

s

0

(

k, ε

)

and for all

a

1

, a

2

, . . . , a

s

suh that

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

₋

₁

_}

itholds

F

n(

J

k(

a

1

, a

2

, . . . , a

s))

V

(

J

k(

a

1

, a

2

, . . . , a

s))

−

1

6

ε,

where

V

(

J

k

(

a

1

, a

2

, . . . , a

s

)) =

p

−

sk

is the

s

-dimensional volume of the mentioned

paral-lelepiped.

Weomit parameters in denotations

N

f

1

,...,f

s

0

(

k, ε

)

,if they are lear fromthe ontext.

Analogously to the ase of

N

f

1

,...,f

s

₍

_{k, ε}

₎

, using the denotation

N

f

1

,...,f

s

0

(

k, ε

) =

g

(

k, ε

)

,

we mean that the given funtion

g

satises onditions imposed on it in Denition 2, but

the minimum of the estimate is not guaranteed.

3 The d-uniformity

In essene, the error mentioned in Denition 1 is absolute for any parallelepiped,

hyperubes; this property implies the followingassertion.

Theorem 2. For a olletion of ompatible funtions

f

1

, f

2

, . . . , f

s

Denition 1 is

equiv-alent to Denition 2, and

N

(

ε

) =

N

0

(

k, ε

′

₎

, where

k

=

−

log

p

(

ε/

4

s

)

, ε

′

= 1

/

2

ε

.

Proof. Letonditions of Denition 2 befullled. Let usprovethat

∀

ε >

0 :

_∃

N

(

ε

)

_∀

n > N,

_∀

J

:

_|

V

(

J

)

₋

F

n(

J

)

_|

< ε.

Choose

k

=

−

log

p

(

ε/

4

s

)

, ε

′

= 1

/

2

ε

. Let us prove that the number

N

=

N

0

(

k, ε

′

₎

is the

desiredone.

Let

J

be some parallelepiped in

[0

,

1)

s

. Wedenote by

J

+

k

the union of hyperubes in

the form

J

k(

a

1

, . . . , a

s)

whih have at least one ommon point with

J

and we do by

J

−

k

the union of hyperubes whih entirelylie inside

J

.

Evidently,

J

+

k

forms aparallelepiped and sodoes

J

−

k

.

Note that

0

6

V

(

J

+

k

)

−

V

(

J

k

−

)

6

ε/

2

. Really, ineah of

s

measurements there exist

nomorethan two ¾layers¿ of the lattiethat liein

J

+

k

,but donot liein

J

−

k

. Thevolume

of eah of them does not exeed

1

/p

k

. Therefore, the total dierene does not exeed

2

s/p

k

₌

_ε/

₂

.

Let usnow write the ondition of Denition 2in the form

V

(

J

k

(

a

1

, . . . , a

s

))(1

−

ε

′

)

6

F

n

(

J

k

(

a

1

, . . . , a

s

))

6

V

(

J

k

(

a

1

, . . . , a

s

))(1 +

ε

′

)

and alulatethe sum foreah of sets

J

+

k

, J

k

−

. We obtain

V

(

J

_k

+

)(1

₋

ε

′

)

6

F

n(

J

_k

+

)

6

V

(

J

_k

+

)(1 +

ε

′

)

V

(

J

_k

−

)(1

₋

ε

′

)

6

F

n(

J

_k

−

)

6

V

(

J

_k

−

)(1 +

ε

′

)

.

It is also evident that

V

(

J

−

k

)

6

V

(

J

)

6

V

(

J

k

+

)

. In addition,

F

n(

J

−

k

)

6

F

(

J

)

6

F

n(

J

k

+

)

,

beause these values are proportional to the number of points of the projetion that lie

inside the orresponding parallelepiped. Then we write

F

n(

J

)

₋

V

(

J

)

6

F

n(

J

_k

+

)

₋

V

(

J

_k

−

)

6

V

(

J

_k

+

)(1 +

ε

′

)

₋

V

(

J

_k

−

)

6

V

(

J

_k

+

)

ε

′

+

ε/

2

.

Sine

J

+

k

lies inside the unit ube, we have

V

(

J

+

k

)

6

1

, and this means that the latter

expression does not exeed

ε

(beause

ε

′

₌

_ε/

₂

). Analogously,

|

F

n(

J

)

₋

V

(

J

)

_|

6

ε

for anarbitrary parallelepiped

J

.

The proof of the onverse assertion is performed with the help of an analogous

esti-mation.

Letusstrengthen Denition2,xing the residue

x

of thedivision by

p

d

fornatural

d

.

Let

d

∈

N

, β

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

. Denote by

P

(

n, d, β

)

, n

>

d

the multiset of points

f

1

(

x

) mod

p

n

p

n

,

f

2

(

x

) mod

p

n

p

n

, . . . ,

f

s(

x

) mod

p

n

p

n

for all

x

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

_{, x}

_mod

_p

d

₌

_β

. Let

F

d,β

n

(

J

)

be the ratio of the number of

points from

P

(

n, d, β

)

that liein

J

tothe ardinal number of

P

(

n, d, β

)

,i.e., to

p

n

−

d

.

Denition 3. The joint projetion of ompatible funtions

f

1

, . . . , f

s

is said to be

d

-uniformlydistributed,if forany number

ε >

0

andforanynatural

k, d

thereexists natural

N

=

N

f

1

,...,f

s

d

(

k, ε

)

suh that for eah

n > N

, for any

β

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

, and for all

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

−

1

}

it holds

F

d,β

n

(

J

k(

a

1

, a

2

, . . . , a

s))

V

(

J

k(

a

1

, a

2

, . . . , a

s))

−

1

6

ε.

In other words, the

d

-uniformity is the uniformity in eah

p

-adi ball, whose volume

equals

p

d

.

For

N

f

1

,...,f

s

d

(

k, ε

)

we use assumptions analogous to above ones for

N

and

N

0

(see the

end part of Setion2).

Note that the number

N

d

indiated in the denition is independent of

β

; it depends

only on its

p

-adi length, i.e., on

d

. This ondition does not strengthen the denition.

Really,sinefor xed

d

thenumberof possiblevalues of

β

isnite,one ouldhavehosen

N

d

as the maximum of the orresponding numbers for eah

β

. Nevertheless, it is more

onvenient touse just this statement, i.e., the uniformitywith respet to

β

.

Evidently, the

d

-uniformityimpliesthe uniformityinthesenseofDenition2,namely,

it sues to put

d

= 0

. In what follows, when speaking of the uniformity, we mean the

d

-uniformity, if this leads tono ambiguity.

Let

n

∈

N

, n

>

k, x

1

, x

2

, . . . , x

s

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

. Note that the point

x

1

p

n

,

x

2

p

n

, . . . ,

x

_p

n

s

belongs to

J

k(

a

1

, a

2

, . . . , a

s)

if and only if the prex of the length

k

in

p

-adi notation of

x

i

onsists of exatly

n

digits, while the

p

-adi notationof

a

i

onsists

of exatly

k

digits independently of the presene of leadingzeros).

One an alsoeasilysee thatthe ondition

x

mod

p

d

₌

_β

is equivalenttothe fat that

the latter

d p

-adi digitsof

x

ontain the notation of

β

.

4 The oin toss

In this setion we prove some orollaries of the

d

-uniformity whih we need in the

indution step in the proofof the main theorem.

Let

d, r

∈

N

, d < r, β

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

. Denote by

Ω(

r, d, β

)

the set of all

x

∈

{

0

,

1

, . . . , p

r

₋

₁

_}

suh that

x

mod

p

d

₌

_β

.

Speaking informally, the

d

-uniformity of the joint projetion of the olletion

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s(

x

)

means that with xed

d, β

and xed suiently large

r

, under the

equiprobable hoie of

x

from

Ω(

r, d, β

)

,the probabilitythat the point

f

1

(

x

) mod

p

r

p

r

,

f

2

(

x

) mod

p

r

p

r

, . . . ,

f

s(

x

) mod

p

r

p

r

belongs to

J

k(

a

1

, a

2

, . . . , a

s)

equals approximately

V

(

J

k(

a

1

, a

2

, . . . , a

s))

. In the following

lemmaweprovethatiffor

x

wehooseasuientlylongstring,thentheprobabilitythat

this event takesplae with at least one

r

an bearbitrarily lose to1.

More formally, let

k

∈

N

, a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

₋

₁

_}

be given. Fix

n

∈

N

, n

>

d, n

>

k

. We say that

x

is suitable, if there exists

r

∈

N

, d

+

k

6

r

6

n

suh that

f

1

(

x

) mod

p

r

p

r

,

f

2

(

x

) mod

p

r

p

r

, . . . ,

f

s(

x

) mod

p

r

p

r

∈

J

k

(

a

1

, a

2

, . . . , a

s

)

.

Lemma 1. Assume that the joint projetion of a olletion of funtions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s

(

x

)

has a

d

-uniform distribution, a value

ε >

0

and numbers

k, d

_∈

N

are given. Then there exists

L

=

L

f

1

,...,f

s

(

k, ε, d

)

suh that for any

n > L

and any

β

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

,

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

₋

₁

_}

the ratio of the number of

suitable

x

to

|

Ω(

n, d, β

)

|

isat least

1

−

ε

.

Proof. The proof of the lemmais based onasimple idea. Letus have a biasedoin suh

that the probabilityof the head is bounded from belowby somenonzero onstant. Then

by tossing the oin suiently many times one an make the probability of getting at

leastone head arbitrarilyloseto1. One¾oin toss¿onsists inobtaininganew value of

n

,namely,

N

d(

k, ε

)

,where

d

isthepreviousvalueof

n

. Theindependeneofoin¾tosses¿

probability of the head is provided by the

d

-uniformity of the olletion

f

1

, . . . , f

s

. Just

inthis lemmaand in the next one we needDenition 3 that strengthens Denition2.

Put

L

(0)

₌

_d

. Construt a sequene

L

(

i

)

_{, i >}

₀

_,

as follows: let

L

(

i

)

be equal to

N

f

1

,...,f

s

L

(

i

−

1)

₊

_k

(

k,

1

/

2)

. Let

γ

i

be the ratio of the number of unsuitable

x

to

|

Ω(

L

(

i

)

_{, d, β}

₎

_|

.

Evidently,

γ

0

6

1

. Letus estimate

γ

i

. Thedenition of the uniformityand the denition

of

L

(

i

)

imply that

F

L

(

i

−

1)

_,β

L

(

i

)

(

J

k(

a

1

, a

2

, . . . , a

s))

>

V

(

J

k(

a

1

, a

2

, . . . , a

s))

/

2 =

1

2

p

sk

for any

β

: 0

6

β

6

p

L

(

i

−

1)

−

1

. Denote the latter numberby

ε

′

(itorresponds tothe probability

ofgettingthe headinone toss). Therefore, for eah

β

∈ {

0

,

1

, . . . , p

L

(

i

−

1)

−

1

_}

there exists

at least

ε

′

_p

L

(

i

)

₋

_L

(

i

−

1)

ways to ll

L

(

i

)

₋

_L

(

i

−

1)

major positions so as to make obtained

x

suitablefor

n

=

L

(

i

)

(independently of theontentof

L

(

i

−

1)

latterpositions). Thismeans

that the ratio of the numberof ways toll these positionsmaking

x

unsuitable does not

exeed

1

−

ε

′

. Evidently, if

x

is unsuitable for

n

=

L

(

i

)

, then it is also unsuitable for all

lesser

n

,inpartiular,for

n

=

L

(

i

−

1)

. Thenumberofwaystollthe latter

L

(

i

−

1)

positions

making

x

unsuitable for

n

=

L

(

i

−

1)

by denition equals

γ

i

−

1

. Therefore, the ombined

share

γ

i

6

(1

−

ε

′

₎

_γ

i

−

1

, beause in order to make

x

unsuitable, one has to ll the minor

L

(

i

−

1)

positions in any of

γ

i

−

1

p

L

(

i

−

1)

ways; for eah of them there exists no more than

(1

₋

ε

′

₎

_p

L

(

i

)

₋

_L

(

i

−

1)

ways to ll the major

L

(

i

)

₋

_L

(

i

−

1)

positions. Sine

γ

0

6

1

, we obtain

γ

i

6

(1

−

ε

′

)

i

. The latterexpression is a geometri progression with the ratio

1

−

ε

′

_<

₁

,

whih meansthat

γ

i

< ε

with

i >

log

1

−

ε

′

ε

. The orrespondingnumber

L

(

i

)

isthe desired

value of

L

, beause, evidently, the ratio of suitable

x

doesnot derease with the growth

of

n

.

For

L

f

1

,...,f

s

(

k, ε, d

)

we also use onditions analogous to those desribed earlier for

N

and

N

0

(see the end part of Setion1).

Note that the proof of the lemma also allows us to alulate

L

f

1

,...,f

s

(

k, ε, d

)

. It sues to onstrut the sequene

L

(0)

₌

_d

,

L

(

i

)

₌

_N

f

1

,...,f

s

L

(

i

−

1)

₊

_k

(

k,

1

/

2)

, i >

0

and to put

L

f

1

,...,f

s

(

k, ε, d

) =

L

(

⌈

log

₁

−

1

/

(2

psk

)

ε

⌉

)

.

Let usgeneralize the latter lemma for the ase of several olletions

a

i

. The previous

lemma guarantees that one an ¾trunate¿ a suiently long sequene

x

(i.e., alulate

modulo

p

r

) so as to make the major

k

positions of

f

i(

x

) mod

p

r

oinide with some

arbitraryxedolletion

a

i

. Nowweare goingtoprovethateven ifwehave

m

olletions

a

(

_i

j

)

_{∈ {}

0

,

1

, . . . , p

k

₋

₁

_}

_,

₁

₆

_i

₆

_s,

₁

₆

_j

₆

_m

, then for eah olletion of numbers there

exists its own

r

(

j

)

, i.e., the way to ¾trunate¿

x

and

f

i

(

x

)

so as to make the major

k

positions among

r

(

j

)

positions of the number

f

i(

x

)

form the number

a

(

j

)

i

. In addition,

we want distint

r

(

j

)

to be not too lose, therefore we additionally impose the ondition

r

(

j

)

₋

_r

(

j

−

1)

_>

_∆

Let

k, d,

∆

∈

N

begiven. Assumealsothatnatural

m

isgiven, aswellas

m

olletions

of

s

numbers suh that for eah

j

, where

1

6

j

6

m

, the olletion of numbers

a

(

j

)

i

∈

{

0

,

1

, . . . , p

k

₋

₁

_}

_,

₁

₆

_i

₆

_s,

is dened. We say that

x

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

isonurrently

suitable, if the followingonditions are fullled:

•

x

issuitablefor

k, d, a

i

=

a

(

j

)

i

foreahxed

j

. Wedenotetheorrespondingnumbers

r

by

r

(

j

)

_.

•

r

(

j

)

_{> r}

(

j

−

1)

₊

_k

_{+ ∆}

for

1

< j

6

m

.

The parameter

∆

has the following sense: it is neessary that neighboring segments of

positionsorrespondingtodegrees of

p

,whosevaluesvaryfrom

r

(

j

)

₋

_k

to

r

(

j

)

₋

₁

,should

not be too lose; namely, we require that their diereneshould exeed some xed

∆

.

Lemma 2. Assume that the joint projetion of a olletion of funtions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s(

x

)

has a

d

-uniform distribution, and numbers

ε

>

0

and

k, d,

∆

, m

_∈

N

are given. Then there exists

L

e

=

L

e

f

1

,...,f

s

(

k, ε, d,

∆

, m

)

suh that for any

n >

L

e

, any

β

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

, and any

m

olletions of

s

numbers

a

(

_i

j

)

_{∈ {}

0

,

1

, . . . , p

k

₋

₁

_}

_,

₁

₆

_i

₆

_s,

₁

₆

_j

₆

_m,

the ratio of the number of

x

, whih are

onurrently suitable for

a

(

j

)

i

, to

|

Ω(

n, d, β

)

|

is at least

1

−

ε

.

Proof. Put

ε

′′

_{= 1}

₋

√

m

1

₋

ε

. Let

L

e

(0)

₌

_d

. Construt a sequene

L

e

(1)

_,

_L

e

(2)

_{, . . . ,}

_L

e

(

s

)

as follows:

L

e

(

j

)

equals

L

f

1

,...,f

s

(

k, ε,

L

e

(

j

−

1)

_{+ ∆)}

whih was obtained in aordane with

Lemma 1. Let

γ

(

j

)

be the ratio of admissible

x

for

n

=

L

e

(

j

)

whih are onurrently

suitablefor

m

=

j

and for

j

rst olletions

a

i

(i.e.,forwhihthereexist

r

(1)

_{, r}

(2)

_{, . . . , r}

(

j

)

satisfying the above orrelations). Then, taking into aount the denition of

L

e

(

i

)

, we

onlude that

γ

(

j

)

_>

_γ

(

j

−

1)

₍₁

₋

_ε

′′

₎

. Hene and from the fat that

γ

(0)

_{= 1}

(sine for

j

= 1

no onditions are imposed on

x

, exept its belonging to

Ω(

n, d, β

)

) it follows that

γ

(

m

)

_>

₍₁

₋

_ε

′

₎

m

_{= 1}

₋

_ε

, whih means that

L

e

=

L

e

(

m

)

is the desiredvalue.

For

L

e

f

1

,...,f

s

(

k, ε, d,

∆

, m

)

wealsomakeassumptionsanalogoustothosedesribedabove for

N

and

N

0

(see the end part of Setion 1).

The proof of Lemma 2 alsoallows us to alulate

L

e

f

1

,...,f

s

(

k, ε, d,

∆

, m

)

. It sues to onstrut the sequene

L

e

(0)

₌

_d

,

L

e

(

j

)

₌

_L

f

1

,...,f

s

(

k, ε,

L

e

(

j

−

1)

_{+ ∆)}

with

j >

0

and to put

e

L

f

1

,...,f

s

(

k, ε, d,

∆

, m

) =

L

e

(

m

)

.

5 Game protools for the uniformity

For proving the uniformity of joint projetions of monomialswe need tosimplify and

by us, are preeded by the existential quantier, while those, whose values are being

hosen by our ompetitor, are preeded by the generality quantier.

Letompatiblefuntions

f

1

(

x

)

, f

2

(

x

)

, . . . , f

s(

x

)

begiven. Consider thefollowinggame

of two players, Goodand Evil. The gamehas the followingsheme:

1. Evil hooses numbers

k, d

∈

N

and

ε

1

, ε

2

>

0

.

2. Good hooses the number

N

e

=

N

e

f

1

,...,f

s

d

(

k, ε

1

, ε

2

)

∈

N

.

3. Evilhoosesnumbers

n >

N , β

e

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

anddrawsonaboard

n

suessive

empty ells (positions); inwhat follows we lleah of themwith a number ranging

from 0 to

p

−

1

. We immediately ll the latter

d

ells with the notation of the

number

β

.

4. Good hooses an arbitrary set of empty positions and desribes some set of

admis-sible ways to ll these positions. The ratio of the ardinal number of the latter

set to the total number of ways toll the hosen positions should be not less than

1

₋

ε

1

. Inotherwords,ifGoodhashosen

l

positions,thenthenumberofadmissible

variantsshould be atleast

(1

−

ε

1

)

p

l

.

5. Evil llspositions hosen atthe previous step inone of admissibleways.

6. Good olorsan arbitraryset of empty positions. Let their numberequal

m

.

7. Evil absolutely arbitrarily lls all empty unolored positions and hooses

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

−

1

}

.

8. Good desribes

(1

−

ε

2

)

p

m

−

sk

ways toll olored positions so that for the number

x

_{∈ {}

0

,

1

, . . . , p

n

₋

₁

_}

, whose

p

-adi notationiswritten onthe board, the point

f

1

(

x

) mod

p

n

p

n

,

f

2

(

x

) mod

p

n

p

n

,

f

3

(

x

) mod

p

n

p

n

, . . . ,

f

s(

x

) mod

p

n

p

n

should belong to

J

k(

a

1

, a

2

, . . . , a

s)

. If Good sueeds in doing this, it is said to be

the winner,otherwise Evilbeomes the winner.

Alongwiththe aboverequirements,thestrategyofGoodshouldbedeterminate. This

meansthat if there are two variantsof the behaviorof Evil suhthat the latter performs

the same ations on several rst steps of the game, Good also must perform the same

have desribed the neessary number of ways to ll them, while Good ould have

inde-pendently onsideredallpossible variantsoflling positionswhose oloringwouldnot be

neessary. Nevertheless, for struturing the proof, it is onvenient to ¾separate roles¿ as

was desribed onsteps 6-7.

For

N

e

f

1

,...,f

s

d

(

k, ε

1

, ε

2

)

∈

N

we make assumptions analogous to those desribed earlier

for

N

and

N

0

(see the end part of Setion 1).

Lemma 3. If Good always wins in the desribed game, then the joint projetion of

f

1

, . . . , f

s

has a uniform distribution, while

N

d(

k, ε

) =

N

e

d(

k, ε

′

_{, ε}

′

₎

, where

ε

′

₌

_ε/

₍₂

_p

sk

₎

.

Proof. Denote

˜

F

_n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s)) =

F

n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

p

n

−

d

.

Therefore,

F

˜

d,β

n

(

J

k

(

a

1

, a

2

, . . . , a

s

)

is the number of points from

P

d,β

n

that belong to

J

k(

a

1

, a

2

, . . . , a

s)

.

Let

k, d, a

1

, a

2

, . . . , a

s

be xed. Rewritethe ondition

F

d,β

n

(

J

k(

a

1

, a

2

, . . . , a

s))

V

(

J

k(

a

1

, a

2

, . . . , a

s))

−

1

6

ε

as

(1

₋

ε

)

p

−

sk

6

F

_n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

6

(1 +

ε

)

p

−

sk

or,alternately,

(1

₋

ε

)

p

n

−

d

−

sk

6

F

˜

_n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

6

(1 +

ε

)

p

n

−

d

−

sk

.

Fix some admissible way to ll

l

positions hosen on Step 4. Calulating the sum of all

possible ways to ll

n

−

l

−

d

−

m

unolored positions on Step 7, we obtain that the

quantity of values of

x

,for whih the orrespondingpointsbelong to

J

k(

a

1

, a

2

, . . . , a

s)

,is

no less than

p

n

−

l

−

d

−

m

_·

₍₁

₋

_ε

2

)

p

m

−

sk

= (1

−

ε

2

)

p

n

−

l

−

d

−

sk

. Now by summing this value

overall admissibleways to ll

l

positionshosen onStep 4, weobtain that their amount

is no less than

(1

−

ε

1

)

p

l

. This means that there exist at least

(1

−

ε

1

)(1

−

ε

2

)

p

n

−

d

−

sk

valuesof

x

,eahofwhihsatisesonditionsdesribedonStep8. Thefulllmentofthese

onditions meansthat apointbelongs to

J

k(

a

1

, a

2

, . . . , a

s)

, whih givesthe inequality

J

k(

a

1

, a

2

, . . . , a

s)

and to

J

k(

a

′

1

, a

′

2

, . . . , a

′

s

)

with

(

a

1

, a

2

, . . . , a

s)

6

= (

a

′

1

, a

′

2

, . . . , a

′

s

)

.

There-fore,

˜

F

_n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

6

p

n

−

d

−

X

(

a

′

1

,...,a

′

s

):(

a

′

i

)

6

=(

a

i

)

˜

F

_n

d,β

(

J

k(

a

₁

′

, a

′

₂

, . . . , a

′

_s

))

6

6

p

n

−

d

₋

(

p

sk

₋

1)(1

₋

ε

1

)(1

−

ε

2

)

p

n

−

d

−

sk

=

= (1

₋

(1

₋

ε

1

)(1

−

ε

2

))

p

n

−

d

+ (1

−

ε

1

)(1

−

ε

2

)

p

n

−

d

−

sk

6

6

(

ε

1

+

ε

2

)

p

n

−

d

+

p

n

−

d

−

sk

Put

ε

1

=

ε

2

= min(1

−

√

1

₋

ε, ε/

2

p

sk

₎

. Thenweget

(1

₋

ε

1

)(1

−

ε

2

)

p

n

−

d

−

sk

>

(1

−

ε

)

p

n

−

d

−

sk

,

and

(

ε

1

+

ε

2

)

p

n

−

d

+

p

n

−

d

−

sk

6

(1 +

ε

)

p

n

−

d

−

sk

.

This means that inequalities

(1

₋

ε

1

)(1

−

ε

2

)

p

n

−

d

−

sk

6

F

˜

n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

,

whih were obtained above,and

˜

F

_n

d,β

(

J

k

(

a

1

, a

2

, . . . , a

s

))

6

(

ε

1

+

ε

2

)

p

n

−

d

+

p

n

−

d

−

sk

lead tothe desiredorrelations

(1

₋

ε

)

p

n

−

d

−

sk

6

F

˜

n

d,β

(

J

k(

a

1

, a

2

, . . . , a

s))

6

(1 +

ε

)

p

n

−

d

−

sk

.

Note that

1

−

√

1

₋

ε > ε/

2

with

ε >

0

. Therefore, with

s, k

∈

N

, p

>

2

it holds

min(1

₋

√

1

₋

ε, ε/

(2

p

sk

_{)) =}

_ε/

₍₂

_p

sk

₎

, whih means that in aordane with the said

above,

N

d(

k, ε

) =

N

e

d(

k, ε/

(2

p

sk

₎

_{, ε/}

₍₂

_p

sk

₎₎

.

Consider the following modiation of the game desribed above:

0. Good xes some natural onstant

c

0

=

c

f

1

,...,f

s

0

.

1. Evil xes

k, d

∈

N

and

ε

1

>

0

.

2. Good hooses the number

N

b

=

N

b

f

1

,...,f

s

3. Evilhoosesnumbers

n >

N , β

b

∈ {

0

,

1

, . . . , p

d

₋

₁

_}

anddrawsonaboard

n

sequential

emptyells-positions,eahofwhihinwhatfollowswillbelledwithanumberfrom

0to

p

−

1

. Thelatter

d

ellsare immediatelylledwith thenotationofthe number

β

.

4. Goodhooses anarbitraryset ofempty positionsanddenes someset ofadmissible

ways tollthese positions. The ratioofthe ardinalnumberofthe latterset tothe

total number of ways toll the hosen positions should be atleast

1

−

ε

1

.

5. Evil llspositions hosen onthe previousstep in one of admissibleways.

6. Good olors an arbitrary set of positions whih are not lled yet. Denote their

number by

m

.

7. Evil absolutelyarbitrarily llsall empty positions exept olored ones and hooses

a

1

, a

2

, . . . , a

s

∈ {

0

,

1

, . . . , p

k

−

1

}

.

8. Gooddenes

p

m

−

sk

ways tolloloredpositionssoastosatisfythefollowing

ondi-tion. Denoteby

x

∈ {

0

,

1

, . . . , p

n

₋

₁

_}

thenumber,whose

p

-adinotationwillbe

writ-ten on the board;

x

1

=

f

1

(

x

) mod

p

n

_{, x}

2

=

f

2

(

x

) mod

p

n

, . . . , x

n

=

f

n(

x

) mod

p

n

.

Denoteby

b

i

numbers, whose

p

-adinotationis theprex ofthe length

k

oftherow

onsisting of

n

symbols of the

p

-adi notation of

x

i

(i.e.,

b

i

=

⌊

x

i

/p

n

−

k

_⌋

). Require

that

b

i

shoulddierfrom

a

i

nomorethan by

c

0

(hereinafter intheondition ¾dier

no more than by

c

0

¿ imposed on

p

-adi numbers of the length

k

, the dierene is

understoodastheminimumoftwodierenes alulatedmodulo

p

k

;inotherwords,

0and

p

k

₋

₁

aretreatedasdierent by1). In thease, whenGoodan dene

p

m

−

sk

ways to ll the olored positions, eah of whih satises the above requirement,

Good issaid to be the winner,otherwise Evilis said to win.

Let us impose one additional requirement to the tehniques proposed by Good;

namely,for two distint olletions

a

1

, a

2

, . . . , a

s

, assuming that the rest ations of

Evilare the same, sets of tehniques proposed by Goodshouldbe non-interseting.

As above, the strategy of Good isassumed tobedeterminate.

For

N

b

f

1

,...,f

s

d

(

k, ε

1

)

we makeassumptionsanalogoustothosemadeabovefor

N

and

N

0

(see the end part of Setion 1).

Lemma 4. In the ase, when Good always wins in the modied game, it an also be the

winner in the initial game, and

N

e

d(

k, ε

1

, ε

2

) =

N

b

d(

k

2

, ε

1

)

, where

k

2

=

k

+ log

p

2

sc

f

1

,...,fs

0

ε

2

winning strategy for Good in the modied game. Let us desribe the strategy for a

mediatorthat uses this orale, whihis winningfor the non-modiedgame.

Assume that on the rst step the orale gives a number

c

0

. After obtaining

k

the

mediatoralulates

k

2

=

k

+ log

p

2

sc

ε

2

0

. Note that

(

p

k

2

−

k

₋

₂

_c

0

)

s

p

(

k

2

−

k

)

s

>

(1

−

ε

2

)

.

Really, for

0

< ε

2

<

1

we have

(1

₋

ε

2

)

1

/s

6

1

−

ε

2

s

,

thereforewith

k

2

=

k

−

log

p

2

ε

sc

2

0

it holds

p

k

2

−

k

₋

₂

_c

0

p

k

2

−

k

s

=

1

₋

2

c

0

p

k

2

−

k

s

=

1

₋

ε

2

s

s

>

1

₋

ε

2

.

Furthermore,themediatorwillonurrentlyuse

(

p

k

2

−

k

₋

₂

_c

0

)

s

oraleswhihimplement

thewinningstrategyforthe modiedgame. Onsteps1-6hesendsthedataobtained from

oralestoEvilanddoesthe datagivenbyEviltooralesunhanged, exeptthe fatthat

on Step 1 he sends to the orales

k

2

instead of

k

. Sine the orales are determinate,

the data produed by them oinide. Having obtained onStep 7 numbers

a

1

, a

2

, . . . , a

s

,

the mediator use them to form olletions

a

1

p

k

2

−

k

₊

_b

1

, a

2

p

k

2

−

k

+

b

2

, . . . , a

s

p

k

2

−

k

+

b

s

for all possible ombinations of

b

i

∈ {

c

0

, c

0

+ 1

, . . . , p

k

2

−

k

₋

_c

0

−

1

}

. Therefore he gets

(

p

k

2

−

k

₋

₂

_c

0

)

s

olletions of numbers, eah of whih belongs to

{

0

,

1

, . . . , p

k

2

₋

₁

_}

. Then

he sendsthese olletionstothe orales as

a

1

, a

2

, . . . , a

s

. Onean easilysee that ifany of

the sent numbers varies nomore than by

c

0

, then their rst

k

digits remain equalinitial

a

1

, a

2

, . . . , a

s

. Therefore, the tehniques for lling

m

olored positions dened by orales

satisfy requirements imposed on Step 8 of the non-modied game. Eah orale denes

p

m

−

k

2

s

tehniques, therefore, their total number is

p

m

−

k

2

s

₍

_p

k

2

−

k

₋

₂

_c

0

)

s

. The additional

requirement desribed on Step 8 guarantees that no tehnique is ounted twie. The

inequality

p

m

−

k

2

s

₍

_p

k

2

−

k

₋

₂

_c

0

)

s

p

m

−

ks

>

p

m

−

k

2

s

₍₁

₋

_ε

2

)(

p

k

2

−

k

)

s

p

m

−

ks

= 1

−

ε

2

ompletes the proof of the lemma.

6 The uniformity of the joint projetion of monomials

Let us rst prove one simple assertionwhih willallow us to restritthe variation of