正特徵域上丟番圖逼近的賦距結果

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行政院國家科學委員會專題研究計畫成果報告

正特徵域上丟番圖逼近的賦距結果

研究成果報告(精簡版)

計畫類別：個別型

計畫編號： NSC 98-2115-M-009-009-

執行期間： 98 年 08 月 01 日至 99 年 07 月 31 日

執行單位：國立交通大學應用數學系（所）

計畫主持人：符麥克

計畫參與人員：大專生-兼任助理人員：唐易格

博士班研究生-兼任助理人員：廖康伶

博士班研究生-兼任助理人員：林光暉

報告附件：出席國際會議研究心得報告及發表論文

處理方式：本計畫可公開查詢

中華民國 99 年 08 月 17 日

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Project: Metrical results for Diophantine approximation in positive

characteristic

by

Michael Fuchs

1 General

This is the final report on the National Science Council project “Metrical results for Diophantine

approximation in positive characteristic” with grant number NSC-98-2115-M-009-009 and term

from August 1st, 2009 to July 31st, 2010.

Before presenting the outcomes of this project, we shortly summarize them.

• All conjectures (as well as generalizations) of the project proposal have been established.

• The two papers [2] and [3] contain the main findings of this project (preprints of the papers

are attached to this report). The first one appeared this year in Acta Arithmetica and the

second one was accepted by the same journal.

2 Results

In order to describe our results, we need some notations. First, let F

q

denote a finite field with q

elements. Moreover, denote by F

q

[T ] the polynomial ring over F

q

and by

F

q

((T

−1

)) =

(

f =

X

i≤n

a

i

T

i

: n ∈ Z, a

i

∈ F

q

, a

n

6= 0

)

∪ {0}

the field of formal Laurent series over F

q

. We define a norm in the usual way as |f | = q

n

if f 6= 0

and |0| = 0. By restricting this norm to the set

L = {f ∈ F

q

((T

−1

)) : |f | < 1}

one obtains a compact Abelian group. Hence, there exists a unique, translation-invariant

probabil-ity measure which we are going to denote by m.

Several recent papers have studied the following Diophantine approximation problem

f −

P

Q

<

1 q

2n+ln

, Q monic, deg Q = n, gcd(P, Q) = 1,

(1)

where f ∈ L is random (with respect to m) and l

n

is a sequence of non-negative integers.

In order to put our results into context, we highlight some recent results. First, in [4] the

following strong law of large number with error term was established.

Theorem 1 (K. Inoue and H. Nakada). The number of solutions of (1) with deg Q ≤ N satisfies

q − 1

q

Ψ(N ) + O (Ψ(N ))

1/2

(log Ψ(N ))

3/2+

a.s.

with an arbitrary

> 0 and Ψ(N ) :=

P

n≥N

q

−ln

_.

(3)

Moreover, in [5] the authors studied (1) with the condition gcd(P, Q) = 1 dropped. They

proved the following result.

Theorem 2 (H. Nakada and R. Natsui). Let l

n

be non-decreasing. Then, under some further

tech-nical conditions on

l

n

, the number of solutions of (1) without the condition

gcd(P, Q) = 1 and

deg Q ≤ N is a.s. asymptotic to Ψ(N ).

Note that compared to the previous result, the conditions in Theorem 2 are more restrictive and

the result is less precise. Our starting point of this project was to improve this result and extend it

to a more general setting.

Inhomogeneous Diophantine Approximation.

Consider the inhomogeneous Diophantine

ap-proximation problem

f −

g + P

Q

<

1 q

2n+ln

, Q monic, deg Q = n,

(2)

where f ∈ L is random, g ∈ L and l

n

is a sequence of non-negative integers.

Using an ingenious method of W. M. Schmidt [6], we proved the following result in [2].

Theorem 3. For any fixed g ∈ L, the number of solutions of (2) with deg Q ≤ N satisfies

Ψ(N ) + O (Ψ(N ))

1/2

(log Ψ(N ))

2+

a.s.

with an arbitrary

> 0.

This result is remarkable because of the following reasons.

• For g = 0, it improves upon Theorem 2 by removing ALL restrictions on l

n

and providing

an error term.

• For g = 0, it completes the result in [1] where Diophantine approximation of linear forms

with at least two terms was studied (our result covers the missing case of only one term).

• The error term is better and the conditions are less restrictive as in the corresponding result

in the real case; see [7].

Then, we also considered (2) with several restrictions on Q.

Restricted Diophantine Approximation.

Here, we proved a variety of results in [2]. We just

state some consequences of our results; for more consequences and general results the reader is

referred to [2].

Theorem 4.

(i) Let

_{C, D ∈ F}

q

[T ] with deg C < deg D. The number of solutions of (2) with

Q ≡ C mod D and deg Q ≤ N satisfies

1 |D|

Ψ(N ) + O (Ψ(N ))

1/2

_{(log Ψ(N ))}

2+

a.s.

(4)

(ii) The number of solutions of (2) with

Q square-free and deg Q ≤ N satisfies

q − 1

q

Ψ(N ) + O (Ψ(N ))

1/2

_{(log Ψ(N ))}

2+

a.s.

with an arbitrary

> 0.

Moreover, if restricting Q to the set of irreducible polynomials, we even have a better error

term.

Theorem 5. Let

Ψ

1

(N ) :=

X

n≤N

1 nq

ln

.

Then, the number of solutions of (2) with

Q irreducible and deg Q ≤ N satisfies

Ψ

1

(N ) + O (Ψ

1

(N ))

1/2

(log Ψ

1

(N ))

3/2+

a.s.

with an arbitrary

> 0.

Simultaneous Diophantine Approximation.

Now, consider the simultaneous Diophantine

ap-proximation problem

f

j

−

P

j

Q

<

1 q

n+l(j)n

, Q monic, deg Q = n, j = 1, . . . , d,

(3)

where (f

1

, . . . , f

d

) ∈ L × · · · × L is random (with respect to the m-fold product measure of m)

and l

(j)n

are sequences of non-negative integers. Moreover, set l

n

:=

P

d

j=1

l

(j) n

.

Using Schmidt’s method once more, we proved the following result which generalizes

Theo-rem 3 above (for g = 0).

Theorem 6. Let l

n

≥ n. Then, the number of solutions of (3) with deg Q ≤ N satisfies

Ψ(N ) + O (Ψ(N ))

1/2

(log Ψ(N ))

2+

a.s.

with an arbitrary

> 0.

Moreover, in [3], we considered (3) with the additional condition gcd(P

j

, Q) = 1.

Theorem 7. Let l

n

≥ n. Then, the number of solutions of (3) with gcd(P

j

, Q) = 1 and deg Q ≤ N

satisfies

c

0

Ψ(N ) + O (Ψ(N ))

1/2+

a.s.

with an arbitrary

> 0. Here,

c

0

:=

X

Q1 monic

· · ·

X

Qdmonic

µ(Q

1

)

|Q

1

|

· · ·

µ(Q

d

)

|Q

d

|

1 |lcm(Q

1

, . . . , Q

d

)|

> 0,

where

µ(·) is the Moebius µ function.

This result generalizes Theorem 1 to the multi-dimensional setting. Note, however, that the

error term in our result for d = 1 is weaker than the error term in Theorem 1. This is due to the

fact that we use a completely different (and more involved) method of proof (the method of proof

of Theorem 1 relied on continued fraction theory which is not available in higher dimensions).

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3 Summary

In this project, we established several new results concerning inhomogeneous Diophantine

ap-proximation, restricted Diophantine approximation and simultaneous Diophantine approximation

in the field of formal Laurent series over a finite base field. In particular, we were able to verify

all conjectures from the project proposal. Moreover, our results improve and generalize several

previous results in this area. Finally, our results hold under less restrictive assumptions and are

more precise compared to the corresponding results over the real number field.

References

[1] M. M. Dodson, S. Kristensen, and J. Levesley (2005). A quantitative Khintchine-Groshev

type theorem over a field of formal series, Indag. Math. (N.S.), 16, 171-177.

[2] M. Fuchs (2010). Metrical theorems for inhomogeneous Diophantine approximation in

pos-itive characteristic, Acta Arith., 141, 191-208.

[3] M. Fuchs. A note on simultaneous Diophantine approximation in positive characertistic, Acta

Arith., accepted.

[4] K. Inoue and H. Nakada (2003). On metric Diophantine approximation in positive

character-istic, Acta ARith., 110, 205-218.

[5] H. Nakada and R. Natsui (2006). Asymptotic behavior of the number of solutions for

non-Archimedean Diophantine approximations, Acta Arith., 125, 203-214.

[6] W. M. Schmidt (1960). A metrical theorem in Diophantine approximation, Canad. J. Math.,

12, 619-631.

[7] W. M. Schmidt (1964). Metrical theorems on fractional parts of sequences, Trans. Amer.

Math. Soc., 110, 493-518.

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Metrical Theorems for Inhomogeneous Diophantine

Approximation in Positive Characteristic

Michael FUCHS

Dedicated to Prof. Harald Niederreiter

on the occasion of his 65th birthday

Abstract

We consider inhomogeneous Diophantine approximation for formal Laurent series over a finite base

field. We establish an analogue of a strong law of large numbers due to W. M. Schmidt with a better

error term than in the real case. A special case of our result improves upon a recent result by H. Nakada

and R. Natsui and completes a result of M. M. Dodson, S. Kristensen, and J. Levesley. Moreover, we

prove various results for inhomogeneous Diophantine approximation with restricted denominators.

1 Introduction

Several recent studies have been concerned with the metric theory of Diophantine approximation in the

field of formal Laurent series; for some references see below. The aim of this paper is to make some

further progress on the inhomogeneous Diophantine approximation problem. More precisely, we will

establish some analogues of results from the real number case (which in the sequel will be referred to

as the ”classical case”) with some improvements which are arising from the more simple nature of the

metric structure of the formal Laurent series field.

First, let us fix some notation. Subsequently, we will denote by F

q

a finite field with q elements; the

polynomial ring over F

q

, the field of rational functions over F

q

, and the field of formal Laurent series

over F

q

will be denoted by F

q

[T ], F

q

(T ), and F

q

((T

−1

)), respectively. For f ∈ F

q

((T

−1

)) with

f = a

n

T

n

+ a

n−1

T

n−1

+ · · · ,

a

k

∈ F

q

, a

n

6= 0, n ∈ Z,

we define |f | := q

n

and |0| := 0. It is easily checked that | · | is a norm which satisfies the ultra-metric

property, i.e.,

|f − g| ≤ max{|f |, |g|}

with equality if |f | 6= |g|. This property in particular implies that two balls (defined in the standard

way) are either disjoint or they are contained in each other. Finally, we set

L = {f ∈ F

q

((T

−1

)) : |f | < 1}.

Note that L equipped with the restriction of the norm to L is a compact abelian group. Consequently,

there exist a unique, translation-invariant probability measure which will be denoted by m.

Key words: formal Laurent series, inhomogeneous Diophantine approximation, Diophantine approximation with restricted

denominators, strong laws of large numbers, Schmidt’s method. 2000 Mathematics Subject Classification: 11J61, 11J83, 11K60.

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In the following, we will be concerned with the inhomogeneous Diophantine approximation

prob-lem: for f, g ∈ L consider the Diophantine inequality

|Qf − g − P | <

1 q

n+ln

, Q is monic, deg Q = n,

(1)

whose solutions are pairs of polynomials hP, Qi ∈ F

q

[T ] × F

q

[T ] with Q 6= 0 (throughout this work

we will use h·, ·i to denote pairs, whereas (·, ·) is reserved for the gcd). Here, l

n

is a sequence of

non-negative integers. In particular, note that l

n

just depends on deg Q.

In a recent paper, C. Ma and W.-Y. Su [

8 ] investigated the above problem and proved a Khintchine

type 0-1 law for the number of solutions if both f and g are chosen randomly (with respect to m) from

L. Their result is an analogue of a result of J. W. S. Cassels [

3 ] from the classical case, where this

situation is sometimes called the ”double-metric” case. Moreover, the following two ”single-metric”

cases were considered over the real number field as well (e.g., see [

11 ] and [

12 ]): (S1) fix f and choose

a random g ∈ L; (S2) fix g and choose a random f ∈ L.

In this paper, we are interested in stochastic properties of the solution set of (

1 ) for f, g such that

the number of solutions is infinite. More precisely, we will derive strong laws of large numbers with

error terms for the number of solutions hP, Qi of (

1 ) with deg Q ≤ N . Such results have so far only

been established for (S2) with g = 0; see [

6 ] and H. Nakada and R. Natsui [

9 ]. Here, we will further

improve these results and extend them to general g. So, the main part of the paper will focus on the

case (S2). The other ”single-metric” case and the ”double metric” case exhibit a somehow different

behavior and will be only briefly discussed in the final section.

From now on, let g ∈ L be fixed. Moreover, define

Ψ(N ) :=

X

n≤N

1 q

ln

.

Our first result reads as follows.

Theorem 1. The number of solutions of (

1 ) with 0 ≤ deg Q ≤ N satisfies

Ψ(N ) + O

Ψ(N )

1/2

(log Ψ(N ))

2+

,

a.s.,

where > 0 is an arbitrary constant.

This result is an analogue of a result of W. M. Schmidt [

11 ] from the classical case. In fact, we will

use a variant of Schmidt’s method to prove it. Note, however, that the error term is better than the one

from the classical case. Moreover, no monotonicity assumption on l

n

is required.

For g = 0 the improved error term was also achieved in the classical case; see G. Harman [

7 ].

The result in this special case improves upon Theorem 3 in [

9 ] by removing some further technical

conditions on l

n

and providing an error term. Moreover, our result completes the main result in [

4 ]

which was concerned with Diophantine approximation of linear forms with at least two terms. Here,

the missing case of only one term is considered. As in the real case, the current situation turns out to

be more complex, a claim which is further supported by the fact that the result in [

4 ] has a better error

term; for a discussion of this phenomena in the real case see [

10 ].

In fact, our method of proof can be used to obtain even more general results. More precisely, the

method will allow us to investigate inhomogeneous Diophantine approximation with restricted

denom-inators as well. Therefore, replace (

1 ) by

|F (Q)f − g − P | <

1 q

n+ln

, Q is monic, deg Q = n,

(2)

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First, we will fix some further notation. Let

F := {Q : Q monic and F (Q) 6= 0}

and denote by F

n

the subset of all polynomials Q ∈ F with deg Q = n. Subsequently, we will

only consider F that satisfy the following property: for Q, Q

0

∈ F with deg Q ≤ deg Q

0

, we have

deg F (Q) ≤ deg F (Q

0

). Finally, set

Ψ(N, F ) :=

X

n≤N

#F

n

q

n+ln

.

Then, the following generalization of the above result holds.

Theorem 2. Assume that F (Q) is either Q or 0. Then, the number of solutions of (

2 ) with Q ∈ F and

0 ≤ deg Q ≤ N satisfies

Ψ(N, F ) + O

(Ψ(N ))

1/2

(log Ψ(N ))

2+

,

a.s.,

(3)

where > 0 is an arbitrary constant.

In particular, the latter result gives a meaningful asymptotic formula whenever

lim inf

n→∞

#F

n

q

n

> 0.

(4)

Two important special cases are collected in the following corollary, the first of which has to be

com-pared with the results in [

6 ].

Corollary 1.

(i) Let C, D ∈ F

q

[T ] with deg C < deg D. Then, the number of solutions of (

1 ) with

Q ≡ C (D) and 0 ≤ deg Q ≤ N satisfies

1 |D|

Ψ(N ) + O

(Ψ(N ))

1/2

(log Ψ(N ))

2+

,

a.s.,

(5)

where > 0 is an arbitrary constant.

(ii) The number of solutions of (

1 ) with Q monic, square-free and 0 ≤ deg Q ≤ N satisfies

q − 1

q

Ψ(N ) + O

(Ψ(N ))

1/2

(log Ψ(N ))

2+

,

a.s.,

(6)

where > 0 is an arbitrary constant.

Note that condition (

4 ) is not satisfied for some interesting F such as the set of monic, irreducible

polynomials. This situation, however, turns out to be more simpler and we can obtain a strong law of

large numbers with an even better error term. Therefore, we first prove an analogue of Theorem 3.1 in

[

7 ] which holds for general F .

Theorem 3. The number of solutions of (

2 ) with Q ∈ F and 0 ≤ deg Q ≤ N satisfies

Ψ(N, F ) + O

(Ψ

0

(N ))

1/2

(log Ψ

0

(N ))

3/2+

,

a.s.,

where > 0 is an arbitrary constant and

Ψ

0

(N ) =

X

n≤N

1 q

n+ln

X

m≤n

X

Q∈Fn

X

Q0_∈F m

|(F (Q), F (Q

0

))|

|F (Q)|

.

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This result entails the following corollary.

Corollary 2.

(i) Let

Ψ

1

(N ) :=

X

n≤N

1 nq

ln

.

Then, the number of solutions of (

1 ) with Q monic, irreducible and 0 ≤ deg Q ≤ N satisfies

Ψ

1

(N ) + O

(Ψ

1

(N ))

1/2

(log Ψ

1

(N ))

3/2+

,

a.s.,

where > 0 is an arbitrary constant.

(ii) Let F (Q) = Q

t

with t ≥ 2. Then, the number of solutions of (

2 ) with 0 ≤ deg Q ≤ N satisfies

Ψ(N ) + O

(Ψ(N ))

1/2

(log Ψ(N ))

3/2+

,

a.s.,

where > 0 is an arbitrary constant.

It is worth mentioning that Theorem

3 does not give a meaningful result in the situations discussed

in Theorem

1 and Corollary

1 . Consequently, part (ii) of Corollary

2 shows that the complexity of t = 1

and t ≥ 2 are rather different.

We conclude the introduction by giving a short plan of the paper. In the next section, we will prove

a weak independence result which will form the crucial step in deriving all results above. In particular,

Theorem

3 will follow rather quickly from this result and this will be demonstrated in the next section

as well. Then, in Section 3, we will show how to amend Schmidt’s method to the current situation to

obtain a proof of Theorem

1 and Theorem

2 . In the final section, we will then briefly discuss the other

”single-metric” case and the ”double-metric” case.

Notation. All logarithms appearing throughout this work will only attain values ≥ 1, i.e., log

_a

x should

be interpreted as max{log

_a

x, 1}. We will use Landau’s notation f (x) = O(g(x)) as well as

Vino-gradov’s notation f (x) g(x) to indicate that there exist a constant C ≥ 0 such that |f (x)| ≤ C|g(x)|

for all x sufficiently large.

2 A weak independence result with applications

We start by proving a technical lemma that constitutes a refinement of Lemma 2.3 in [

2 ].

Lemma 1. Let Q, Q

0

be two non-zero polynomials with n = deg Q, m = deg Q

0

and d = deg(Q, Q

0

).

Let l be a non-negative integer. Then, the number N of pairs hP, P

0

i with deg P < n, deg P

0

_{< m and}

g + P

Q

−

g + P

0

Q

0

<

1 q

m+l

(7)

is given by

N

(

= q

n−l

,

if n ≥ l + d;

≤ q

d

_,

_{if n < l + d.}

Proof. First, (

7 ) can be reformulated to

|g(Q

0

− Q) + P Q

0

− P

0

Q| < q

n−l

.

Next, set Q = (Q, Q

0

) · ¯

Q and Q

0

= (Q, Q

0

) · ¯

Q

0

. Then,

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Let −C denote the polynomial part of g( ¯

Q

0

− ¯

Q). Now, we will consider two cases.

First, assume that n < l + d. Then, a necessary condition for hP, P

0

i being a solution of the above

inequality is P ¯

Q

0

− P

0

Q = C. Observe that for P with deg P < n and

¯

P ¯

Q

0

≡ C mod ¯

Q,

(8)

we have P ¯

Q

0

= C + P

0

Q with some polynomial P

¯

0

and

deg P

0

+ deg ¯

Q = deg(P ¯

Q

0

− C) ≤ deg P + deg ¯

Q

0

< n + deg ¯

Q

0

.

Consequently, deg P

0

< m. So, either N = 0 or N equals the number of solutions of (

8 ) which is q

d

.

Next, we consider n ≥ l + d. Here, we can argue similar as above, the only difference being

that N equals the number of solutions of (

8 ) with C replaced by C + D for all polynomials D with

deg D < n − l − d. Consequently, N = q

n−l

_.

Next, we define for Q ∈ F

n

the set

F

Q

:= {f ∈ L : f satisfies (

2 ) with some P ∈ F

q

[T ]}.

Obviously, F

Q

is the union of |F (Q)| disjoint balls. Consequently,

m(F

Q

) =

1 q

n+ln

.

Moreover, we have the following weak independence result.

Proposition 1. Let Q ∈ F

n

, Q

0

∈ F

m

, and d = deg(F (Q), F (Q

0

)). Then,

m(F

Q

∩ F

Q0

) ≤ m(F

_Q

)m(F

_Q0

) + q

d−deg F (Q)−n−ln

.

Proof. First assume that n + l

n

+ deg F (Q) ≥ m + l

m

+ deg F (Q

0

). Then, all balls which make up

F

Q

have radius at most as large as the radius of the balls which make up F

Q0

. So, by the ultra-metric

property of the norm, we have to count how many of the (g + P )/F (Q) are contained in balls with

center (g + P

0

)/F (Q

0

) and radius q

− deg F (Q0)−m−lm

_{, i.e., we have to count the number of solutions of}

g + P

F (Q)

−

g + P

0

F (Q

0

₎

<

1 q

deg F (Q0_)+m+l m

.

The latter number is given by the above lemma. We first consider the case with deg F (Q) ≥ m+l

m

+d.

Here, the number of solutions equals q

deg F (Q)−m−lm

_{. So, we obtain}

m(F

Q

∩ F

Q0

) =

|F (Q)|q

−m−lm

|F (Q)|q

n+ln

=

1 q

n+ln

·

1 q

m+lm

= m(F

Q

)m(F

Q 0

).

Hence, the assertion holds in this case. Now, consider the second case where deg F (Q) < m + l

m

+ d.

Then, again by the above lemma,

m(F

Q

∩ F

Q0

) ≤

q

d

q

deg F (Q)+n+ln

.

Hence, the claim is proved in this case as well.

Next, if n+l

n

deg F (Q) < m+l

m

+deg F (Q

0

), we obtain from the arguments above the claim with

the second term replaced by q

d−deg F (Q0)−m−lm

_{. This term is trivially bounded by q}

d−deg F (Q)−n−ln

_.

Hence, the proof of the proposition is finished.

The above proposition will turn out to be one of the key ingredients in the prove of our results. The

other key ingredient is the following important lemma which is a standard tool in metric number theory.

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Lemma 2 (Lemma 1.5 in [

7 ]). Let ξ

n

(ω) be a sequence of non-negative random variables defined on

a probability space (Ω, B, P ). Let ψ

n

and ϕ

n

be sequences of real numbers with

0 ≤ ψ

n

≤ ϕ

n

.

Define

Φ(N ) =

X

n≤N

ϕ

n

and assume that Φ(N ) → ∞ as N → ∞. Finally, assume that

E





X

M ≤n≤N

ξ

n

− ψ

n





2

X

M ≤n≤N

ϕ

n

.

for all non-negative M < N . Then,

X

n≤N

ξ

n

(ω) =

X

n≤N

ψ

n

+ O

(Φ(N ))

1/2

(log Φ(N ))

3/2+

+ max

n≤N

ψ

n

,

a.s.,

where > 0 is an arbitrary constant.

As a first application of this lemma, we show how to deduce Theorem

3 from it. Therefore, set

ξ

n

:= #{hP, Qi : hP, Qi is a solution of (

2 )}.

This sequence of random variables satisfies the following properties.

Proposition 2.

(i) We have,

E





X

n≤N

ξ

n





= Ψ(N, F ).

(ii) We have,

E





X

M ≤n≤N

ξ

n

−

#F

n

q

n+ln





2

X

M ≤n≤N

1 q

n+ln

X

m≤n

X

Q∈Fn

X

Q0_∈F m

|(F (Q), F (Q

0

))|

|F (Q)|

for all non-negative integers M < N .

Proof. Part (i) follows from

ξ

n

=

X

Q∈Fn

1

FQ

and basic properties of the mean value.

For part (ii), we also use the above representation which yields

E





X

M ≤n≤N

ξ

n

−

#F

n

q

n+ln





2

= 2

X

M ≤n≤N

X

M ≤m≤n−1

X

Q∈Fn,Q0∈Fm

m(F

Q

∩ F

Q0

) − m(F

_Q

)m(F

_Q0

)

+

X

M ≤n≤N

X

Q∈Fn,Q0∈Fm

m(F

Q

∩ F

Q0

) − m(F

_Q

)m(F

_Q0

).

(12)

Now, we can prove Theorem

3 .

Proof of Theorem

3 . If Ψ(N, F ) → c ≥ 0 as N → ∞, the result follows by a standard application of

the Lemma of Borel-Cantelli. Hence, we can assume that Ψ(N, F ) → ∞ as N → ∞. But then the

claim follows from the Proposition above together with Lemma

2 .

Corollary

2 follows from the last result as follows.

Proof of Corollary

2 . For part (i), we use the well-known result (see Chapter 3 in [

1 ])

#F

n

=

q

n

+ O (q

n

_{) ,}

₍₉₎

where < 1 is a suitable constant. Hence,

Ψ(N, F ) = Ψ

1

(N ) + O(1).

Moreover,

Ψ

0

(N ) =

X

n≤N

1 q

2n+ln

X

m≤n

X

deg Q=n Q monic, irreducible

X

deg Q0_=m Q0monic, irreducible

|(Q, Q

0

)| Ψ

1

(N ),

where the last line again follows by (

9 ). This proves the claim.

As for part (ii), first observe that #F

n

= q

n

and hence Ψ(N, F ) = Ψ(N ). The bound for Ψ

0

(N )

is slightly more tricky. First,

Ψ

0

(N ) =

X

n≤N

1 q

(t+1)n+ln

X

m≤n

X

deg Q=n Q monic

X

deg Q0=m Q0_monic

|(Q

t

, (Q

0

)

t

)|

X

n≤N

1 q

(t+1)n+ln

X

deg Q=n Q monic

X

D|Q D monic

q

n

|D|

t

_.

Next, we have

X

deg Q=n Q monic

X

D|Q D monic

|D|

t−1

=

X

d≤n

X

deg D=d D monic

q

n

|D|

t−1

_{= q}

n

X

d≤n

q

(t−1)d

q

tn

.

Plugging this into the estimate above yields Ψ

0

(N ) Ψ(N ). Hence, the result is established.

3 Schmidt’s method in positive characteristic

Note that the method from the last section does not yield a meaningful result for the case F (Q) = Q.

More specifically, it is easily checked that the error term from the proof of part (ii) of Corollary

2 for

t = 1 would be larger than the main term. The same phenomena also occurs in the real case, where this

problem was overcome by an ingenious method introduced by W. M. Schmidt in [

10 ] and [

11 ]. In this

section, Schmidt’s method will be amended to the current situation.

We start with a couple of (easy) lemmas.

Lemma 3 (Dirichlet’s principle in positive characteristic). For all non-zero polynomials Q there exist

polynomials A, B with 0 < |A| ≤ |Q| and (A, B) = 1 such that

g −

B

A

<

1 |A||Q|

.

(13)

Proof. This is proved as in the classical case.

Observe that A and B in the previous lemma just depend on deg Q. Subsequently, for any given

non-zero polynomial Q, we will choose a fixed pair hA, Bi satisfying the assumption of the previous

lemma for a polynomial Q

0

with deg Q

0

= bdeg Q/2c.

Next, we define the following two sets

S(Q; k) = {P : deg P < deg Q and deg(P, Q) ≤ k},

S

∗

(Q; k) = {P : deg P < deg Q and deg(AP + B, Q) ≤ k},

whose cardinalities will be denote by ϕ(Q; k) and ϕ

∗

(Q; k), respectively.

Lemma 4. We have,

ϕ

∗

(Q; k) ≥ ϕ(Q; k).

Proof. First, let Q = Q

1

Q

2

, where every prime factor of Q

1

is also a prime factor of A and (Q

2

, A) = 1.

Then, we have

ϕ(Q; k) ≤ ϕ(Q

1

; k)ϕ(Q

2

; k) ≤ |Q

1

|ϕ(Q

2

; k).

Now, note that AP + B with deg P < deg Q

2

are all different module Q

2

. Hence, ϕ(Q

2

; k) = #{P :

deg P < deg Q

2

and deg(AP + B, Q

2

) ≤ k}. Finally notice that

(AP + B, Q

2

) = (AP + B, Q

1

Q

2

) = (AP + B, Q).

Consequently,

ϕ

∗

(Q; k) = |Q

1

| · #{P : deg P < deg Q

2

and deg(AP + B, Q

2

) ≤ k}.

Combining everything yields the claimed result.

Next, we fix F (Q) = Q. Moreover, as in the last section, it suffices to consider the case where

Ψ(N ) → ∞ as N → ∞. The method of the last section did not work when directly applied to the

sequence ξ

n

. Therefore, we will approximate this sequence by the following one

ξ

_n∗

:= #{hP, Qi : P ∈ S

∗

(Q; Γ(n)) and hP, Qi is a solution of (

1 )},

where Γ(n) = blog

_q

Ψ(n)

2

c. Moreover, similar as in the last section, we define

F

_Q∗

:= {f ∈ L : f satisfies (

1 ) with some P ∈ S

∗

(Q; Γ(n))}.

Then,

ξ

_n∗

=

X

deg Q=n Q monic

1

F∗ Q

and consequently

Eξ

n∗

=

X

deg Q=n Q monic

ϕ

∗

(Q; Γ(n))

q

2n+ln

.

The next result shows that the mean values of the partial sums of ξ

n

and ξ

∗n

are very close to each

other.

Proposition 3. We have,

E





X

M ≤n≤N

ξ

_n∗





=

X

M ≤n≤N

1 q

ln

+ O(1)

for all non-negative integers M < N .

(14)

Proof. First, observe that

0 ≤

X

M ≤n≤N

1 q

ln

− E





X

M ≤n≤N

ξ

_n∗





=

X

M ≤n≤N

X

deg Q=n Q monic

q

n

− ϕ

∗

_{(Q; Γ(n))}

q

2n+ln

≤

X

M ≤n≤N

X

deg Q=n Q monic

q

n

− ϕ(Q; Γ(n))

q

2n+ln

,

where we have used the above lemma in the last step. Next, it is well-known (see [

5 ]) that the number

of pairs hP, Qi with deg P = l < deg Q = n, P, Q monic and deg(P, Q) = k < l is given by

q

n+l−k

1 −

1 q

.

Consequently,

X

deg Q=n Q monic

ϕ(Q, Γ(n)) =

(q − 1)

2

q

n−1

X

l=Γ(n)+1 Γ(n)

X

k=0

q

n+l−k

+ O





Γ(n)

X

l=0 l

X

k=0

q

n+l−k





= q

2n

+ O

q

2n−Γ(n)

.

Plugging this into the above expression, we obtain

0 ≤

X

M ≤n≤N

1 q

ln

− E





X

M ≤n≤N

ξ

_n∗





X

N ≤n≤M

1 q

ln

_Ψ(n)

2

.

Since the latter series is convergent by the Abel-Dini theorem, the claim is proved.

Finally, we need the following property.

Proposition 4. We have,

E





X

M ≤n≤N

ξ

_n∗

−

1 q

ln





2

X

M ≤n≤N

Γ(n)

q

ln

for all non-negative integers M < N .

Proof. We start with an observation that is needed below. By a close inspection of the proof of

Propo-sition

1 , we have

m(F

_Q∗

∩ F

_Q∗0

) ≤

1 q

n+ln

·

1 q

m+lm

+

1 q

2n+ln

A(Q, Q

0

),

(10)

where A(Q, Q) is the number of all pairs P, P

0

with P ∈ S

∗

(Q; Γ(n)), P

0

∈ S

∗

(Q

0

; Γ(m)) and

|g(Q − Q

0

) + P

0

Q − P Q

0

| < min

n

|(Q, Q

0

)|, q

max{n−m−lm,m−n−ln}

o

.

(11)

Moreover, observe that A(Q, Q) ≤ |(Q, Q

0

)|.

We will use this to bound the expected value from the claim. First,

E

X

M ≤n≤N

ξ

∗_n

−

1 q

ln

!

2

=

(15)

=

X

M ≤n≤N

X

M ≤m≤N

Eξ

n∗

· ξ

m∗

− 2

X

M ≤n≤N

1 q

ln

E





X

N ≤n≤M

ξ

_n∗





+

X

M ≤n≤N

X

M ≤m≤N

1 q

ln

·

1 q

lm

=

X

M ≤n≤N

X

M ≤m≤N

Eξ

n∗

· ξ

∗ m

−

1 q

ln

·

1 q

lm

+ O





X

M ≤n≤N

1 q

ln





= 2

X

M ≤n≤N

X

M ≤m≤n−1

Eξ

∗n

· ξ

∗ m

−

1 q

ln

·

1 q

lm

+

X

M ≤n≤N

E(ξ

∗n

)

2

−

1 q

2ln

+ O





X

M ≤n≤N

1 q

ln





,

where the third step follows from Proposition

3 . Now, applying (

10 ) gives

X

M ≤m≤n

Eξ

n∗

· ξ

m∗

=

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q=m Q monic

m(F

_Q∗

∩ F

_Q∗0

)

≤

1 q

ln

· X

M ≤m≤n

1 q

lm

+

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0=m Q0monic

A(Q, Q

0

)

Using this to bound the first and second term in the expression above yields

E





X

M ≤n≤N

ξ

_n∗

−

1 q

ln





2

X

M ≤n≤N

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0=m Q0_monic

A(Q, Q

0

)+

X

M ≤n≤N

1 q

ln

. (12)

Next, we will estimate

Σ :=

X

M ≤n≤N

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0_=m Q0_monic

A(Q, Q

0

).

Therefore, we fix an arbitrary small δ and break Σ into two parts Σ

0

and Σ

00

, where the first part runs

over all pairs hQ, Q

0

i with deg Q

0

_{≤ dn − δ deg(Q, Q}

0

_{)e and the second part runs over the remaining}

pairs. In order to bound Σ

0

, we change the order of summation as follows: first we sum over Q, then

over D|Q and finally over Q

0

with D = (Q, Q

0

). Note that for fixed Q and D the number of Q

0

’s is

bounded by q

n

/|D|

1+δ

. This together with A(Q, Q

0

) ≤ |D| then yields

Σ

0

=

X

M ≤n≤N

1 q

2n+ln

X

deg Q=n Q monic

X

D|Q D monic

q

n

|D|

1+δ

|D|

X

M ≤n≤N

1 q

ln

X

deg D≤n D monic

1 |D|

1+δ

X

M ≤n≤N

1 q

ln

.

As for Σ

00

observe that deg Q

0

> dn − δ deg(Q, Q

0

)e implies

min

n

|(Q, Q

0

)|, q

max{n−m−lm,m−n−ln}

o

_{< |(Q, Q}

0

_)|

δ

_.

Hence, for all hQ, Q

0

i involved in the range of Σ

00

_{the relation (}

₁₁

_{) can be replaced by}

|g(Q − Q

0

) + P

0

Q − P Q

0

| < |(Q, Q

0

)|

δ

.

(13)

This yields

Σ

00

X

M ≤n≤N

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0=m Q0monic

B(Q, Q

0

),

(16)

where B(Q, Q

0

) denotes the number of all P, P

0

with P ∈ S

∗

(Q; Γ(n)) and P

0

∈ S

∗

_(Q

0

_{; Γ(m)) that}

satisfy (

13 ). Again note that B(Q, Q

0

) ≤ |(Q, Q

0

)|.

Collecting all bounds so far, we see that the right hand side of (

12 ) can be replaced by

X

M ≤n≤N

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0_=m Q0_monic

B(Q, Q

0

) +

X

M ≤n≤N

1 q

ln

.

(14)

Next, we will estimate the first term

Σ

0

:=

X

M ≤n≤N

1 q

2n+ln

X

M ≤m≤n

X

deg Q=n Q monic

X

deg Q0=m Q0monic

B(Q, Q

0

)

which we will break into three parts Σ

0₀

, Σ

00₀

, Σ

000₀

, where the ranges will be given below. For every part

we will proceed similar as for Σ

0

above. More precisely, we will change the order of summation as

follows: as for Σ

0

the first two sums will run over Q and D|Q. The final sum will run over ¯

Q

0

with

( ¯

Q

0

, Q/D) = 1. Here, we introduce the notation Q

0

= D ¯

Q

0

and Q = D ¯

Q. Using this notation, we can

rewrite (

13 ) to

|g( ¯

Q − ¯

Q

0

) + P

0

Q − P ¯

¯

Q

0

| < |D|

−1+δ

.

(15)

Finally, we need the notation R = g − B/A, where hA, Bi is the pair belonging to Q. Now, we will

separately estimate the three parts Σ

0₀

, Σ

00₀

, Σ

000₀

.

As for Σ

0₀

, the first two sums of this part run over all hQ, Di with D|Q and |A| ≥ |D|

δ1

_{, where δ}

1

will be chosen later. The last sum runs over ¯

Q

0

and our goal is to count the number of ¯

Q

0

such that (

15 )

has solutions in P, P

0

(whose number will then be bounded by |D|). First, we consider ¯

Q

0

of the form

¯

Q

0

= C

1

+ C

2

, where C

1

is fixed and C

2

is an arbitrary polynomial with deg C

2

< deg A. Plugging

this into (

15 ) and doing some simplifications yields

|gC

2

+ L + ¯

g| < |D|

−1+δ

,

where ¯

g ∈ L does not depend on C

2

∈ F

q

[T ] might depend on C

2

. From the ultra-metric property of

the norm, we obtain

B

A

C

2

+ L + ¯

g

≤ max{|gC

₂

+ L + ¯

g|, |RC

2

|} < max{|D|

−1+δ

, |RA|}.

Observe that since C

2

runs through a complete set of residues modulo A and (A, B) = 1, BC

2

also

runs through a complete set of residues modulo A. Consequently,

C

A

+ ¯

L + ¯

g

< max{|D|

−1+δ

, |RA|},

where we now have to count the number of C’s satisfying this inequality with deg C < deg A. Here,

¯

L is another polynomial that might depend on C. However, since the right hand side of the above

inequality is smaller than 1, ¯

L must be equal to 0. Thus,

|C + A¯

g| < max{|A||D|

−1+δ

, |RA

2

|} ≤ max{|A||D|

−1+δ

, 1}

and the number of such C’s is clearly bounded by |A||D|

−1+δ

+ 1. Next, observe that the number of

C

1

’s above is bounded by |Q||DA|

−1

+ 1. Therefore, the number of ¯

Q

0

such that (

15 ) has a solution in

P, P

0

is bounded by

(|A||D|

−1+δ

+ 1)(|Q||DA|

−1

+ 1) ≤ |Q||D|

−2+δ

+ |Q||D|

−1−δ1

₊

p|Q||D|

−1+δ

_{+ 1}

(17)

where δ

1

, δ are chosen such that δ + δ

1

≤ 1/2. Overall, this yields the following bound for Σ

00

Σ

0₀

X

M ≤n≤N

1 q

2n+ln

X

deg Q=n Q monic

X

D|Q D monic

q

n

|D|

1+δ1

+ 1

|D|

X

M ≤n≤N

1 q

ln

+

X

M ≤n≤N

1 q

n+ln

X

deg D≤n D monic

1 X

M ≤n≤N

1 q

ln

.

(16)

Next, we turn to Σ

00₀

whose first two sums run over all pairs hQ, Di with D|Q, |A| < |D|

δ1

_{, and}

|R| ≥ |D|/|QA|. Again, we will estimate the number of solutions of (

15 ) in ¯

Q

0

, P, P

0

. Therefore, first

observe that (

15 ) can be rewritten as

RC +

L

A

< |D|

−1+δ

(17)

for some polynomials C and L. If L is fixed, then the number of solutions in C of the above inequality

is bounded by |R|

−1

|D|

−1+δ

+ 1. On the other hand, we have

|L| ≤ max{|A||D|

−1+δ

, |RCA| ≤ max{|A||D|

−1+δ

, |RQA|/|D|}.

So, overall, we obtain for the number of C’s such that there exist L satisfying (

17 )

(|R|

−1

|D|

−1+δ

+ 1)(|A||D|

−1+δ

+ |RQA|/|D| + 1)

|QA

2

_||D|

−3+2δ

_{+ |QA||D|}

−2+δ

₊

_p|Q||D|

−1

_{+ 1}

|Q||D|

−2+δ+δ1

₊

p|Q||D|

−1

_{+ 1.}

Note that the above number also equals the number of ¯

Q

0

’s such that (

14 ) has solutions in P, P

0

. Hence,

Σ

00₀

is bounded as follows

Σ

00₀

X

M ≤n≤N

1 q

2n+ln

X

deg Q=n Q monic

X

D|Q D monic

q

n

|D|

2−δ−δ1

+

q

n/2

|D|

+ 1

!

|D|

X

M ≤n≤N

1 q

ln

+

X

M ≤n≤N

1 q

n/2+ln

X

deg D≤n D monic

1 |D|

X

M ≤n≤N

1 q

ln

+

X

M ≤n≤N

n

q

n/2+ln

X

M ≤n≤N

1 q

ln

.

(18)

So, what is left is to bound Σ

000₀

. Here, the first two sums run over all pairs hQ, Di with D|Q, |A| <

|D|

δ1

_{, and |R| < |D|/|QA|. Then, (}

₁₅

_{) together with the ultra-metric property of the norm yields}

| ¯

Q(AP

0

+ B) − ¯

Q

0

(AP + B)| ≤ max{|R( ¯

Q − ¯

Q

0

)A|, |A||g( ¯

Q − ¯

Q

0

) + P

0

Q − P ¯

¯

Q

0

|} < 1.

Consequently,

¯

Q(AP

0

+ B) = ¯

Q

0

(AP + B).

Thus AP + B ≡ 0 ( ¯

Q) and this implies deg ¯

Q ≤ Γ(n). The latter in turn yields deg D ≥ n − Γ(n).

So, in this case, we obtain the bound

Σ

000₀

X

M ≤n≤N

1 q

2n+ln

X

deg Q=n Q monic

X

D|Q,Q monic deg D≥n−Γ(n)

q

n

|D|

=

X

M ≤n≤N

1 q

n+ln

X

deg Q=n Q monic

X

D|Q,Q monic deg D≤Γ(n)

1 X

M ≤n≤N

Γ(n)

q

ln

.

(19)

(18)

Finally, combining (

16 ), (

18 ), and (

19 ) gives the bound

Σ

0

X

M ≤n≤N

Γ(n)

q

ln

.

Plugging this into (

14 ) then proves the claimed result.

Now, we can start with the proof of Theorem

1 .

Proof of Theorem

1 . First, from Proposition

4 together with Lemma

2 , we obtain

X

n≤N

ξ

_n∗

= Ψ(N ) + O

(Ψ

∗

(N ))

1/2

(log Ψ

∗

(N ))

3/2+

,

a.s.,

where > 0 is an arbitrary constant. Next, observe

Ψ

∗

(N ) =

X

n≤N

Γ(n)

q

ln

Ψ(N ) log Ψ(N ).

Hence, the claimed result holds for the sequence ξ

_n∗

.

In order to show that the claimed result holds for ξ

n

as well, observe that from Proposition

3 P





X

n≤N

(ξ

n

− ξ

n∗

) > log Ψ(N )





(log Ψ(N ))

−1

.

Next, choose N

k

to be the minimal positive integer with log Ψ(N

k

) ≥ 2

k

. Then, the Borel-Cantelli

lemma implies that

X

n≤Nk

(ξ

n

− ξ

n∗

) ≤ log Ψ(N

k

)

for almost all f and k large enough. Now, let N be a large enough integer with N

k

≤ N < N

k+1

.

Then,

X

n≤N

(ξ

n

− ξ

n∗

) ≤

X

n≤Nk+1

(ξ

n

− ξ

∗n

) ≤ log Ψ(N

k+1

) log Ψ(N

k

) log Ψ(N ).

Overall, we have shown that for almost all f

X

n≤N

ξ

n

=

X

n≤N

ξ

_n∗

+ O(log Ψ(N )).

Combining with the above result yields the claim.

We note that Theorem

2 also follows from the method above with only minor modifications. So,

what is left is the proof of Corollary

1 .

Proof of Corollary

1 . For part (i), choose F such that

F = {C + LD : monic and L ∈ F

q

[T ]}.

Then, #F

n

= q

n

/|D| for all n ≥ deg D. Consequently,

Ψ(N, F ) =

1 |D|

Ψ(N ) + O(1).

For part (ii), it suffices to point out that it is well-known (see Chapter 3 in [

1 ]) that the number of

monic, square-free polynomials of degree n ≥ 2 is given by q

n

− q

n−1

_{. Hence,}

Ψ(N, F ) =

q − 1

q

Ψ(N ) + O(1).

(19)

4 The ”double-metric” and the other ”single-metric” case

We first turn our attention to the ”double-metric” case. So, in the following, we consider (

1 ) with both

f, g random. As before, we define the set

F

Q

:= {hf, gi ∈ L × L : hf, gi is a solution of (

1 ) with some P ∈ F

q

[T ]},

where Q is a non-zero polynomial.

As already mentioned in the introduction, this case is much easier than the ”single-metric” case

discussed in the previous sections. The reason for this is the second property of the following lemma

which was proved in [

8 ].

Lemma 5.

(i) We have,

(m × m)(F

Q

) =

1 q

n+ln

.

(ii) For Q 6= Q

0

, we have

(m × m)(F

Q

∩ F

Q0

) = (m × m)(F

_Q

)(m × m)(F

_Q0

).

So, if we define

ξ

n

:= #{hP, Qi : hP, Qi is a solution of (

1 )},

then we again have

ξ

n

=

X

deg Q=n Q monic

1

FQ

.

However, the above lemma shows that ξ

n

considered as a sequence of random variables on the product

probability space is pairwise independent. This yields

E





X

M ≤n≤N

ξ

n

−

1 q

ln





2

=

X

M ≤n≤N

Var(ξ

n

) =

X

M ≤n≤N

1 q

ln

1 −

1 q

n+ln

=

X

M ≤n≤N

1 q

ln

+ O(1).

Hence, if we assume that

Ψ(N ) :=

X

n≤N

1 q

ln

→ ∞,

as N → ∞,

then Lemma

2 directly applies and yields the following result (whose proof in case the above

assump-tion does not hold is trivial).

Theorem 4. The number of solutions of (

1 ) with 0 ≤ deg Q ≤ N satisfies

Ψ(N ) + O

(Ψ(N ))

1/2

(log Ψ(N ))

3/2+

,

a.s.,

where > 0 is an arbitrary constant.

Note that a.s. here means with respect to the product measure m × m.

Finally, we briefly discuss the other ”single-metric” case where the roles of f and g are

inter-changed. Therefore, assume now that f is fixed and g is random. Here, without proof, we state the

following result: for any sequence l

n

tending to infinity arbitrarily slowly, there exists an f ∈ L such

that for almost all g the number of solutions of (

1 ) is finite (see P. Sz¨usz [

12 ] for the corresponding

re-sult in the real number case). Consequently, rere-sults of a similar type as in the cases above are impossible

in this case.

(20)

Acknowledgments

Parts of this work were carried out when the author visited the Department of Mathematics, Keio

University. The author wants to thank the department for hospitality and support. Moreover, the author

acknowledges partial support by National Science Council under the grant NSC-98-2115-M-009-009.

References

[1] E. R. Berlekamp (1968). Algebraic coding theory, Mc Graw-Hill.

[2] V. Berth´e, H. Nakada, and R. Natsui (2008). Asymptotic behavior of the number of solutions for

non-Archimedean Diophantine approximations with restricted denominators, Finite Fields Appl.,

14, 849-866.

[3] J. W. S. Cassels (1951). An Introduction to Diophantine Approximation, Cambridge Univ. Press.

[4] M. M. Dodson, S. Kristensen, and J. Levesley (2005). A quantitative Khintchine-Groshev type

theorem over a field of formal series, Indag. Math., 16, 171-177.

[5] M. Drmota and D. Panario (2002). A rigorous proof of the Waterloo algorithm for the discrete

logarithm problem, Des. Codes Cryptography, 26, 229-241.

[6] M. Fuchs (2003). An analogue of a theorem of Sz¨usz for formal Laurent series over finite fields,

J. Number Theory, 101, 105-130.

[7] G. Harman (1998). Metric Number Theory, London Mathematical Society Monographs (New

Series) 18, The Clarendon Press Oxford University Press, New York.

[8] C. Ma and W.-Y. Su (2008). Inhomogeneous Diophantine approximation over the field of formal

Laurent series, Finite Fields Appl., 14, 361-378.

[9] H. Nakada and R. Natsui (2006). Asymptotic behavior of the number of solutions for

non-Archimdedian Diophantine approximations, Acta Arith., 125, 203-214.

[10] W. M. Schmidt (1960). A metrical theorem in Diophantine approximation, Canad. J. Math., 12,

619-631.

[11] W. M. Schmidt (1964). Metrical theorems on fractional parts of sequences, Trans. Amer. Math.

Soc., 110, 493-518.

[12] P. Sz¨usz (1958). ¨

Uber die metrische Theorie der Diophantischen Approximation, Acta Math.

Hun-gar., 9, 177-193.

Michael Fuchs

Department of Applied Mathematics

National Chiao Tung University

1001 Ta Hsue Road

Hsinchu, 300, Taiwan

(21)

A Note on Simultaneous Diophantine Approximation in

Positive Characteristic

Michael Fuchs

Abstract

In a recent paper, Inoue and Nakada proved a 0-1 law and a strong law of large numbers with error

term for the number of coprime solutions of the one-dimensional Diophantine approximation problem

in the field of formal Laurent series over a finite base field. In this note, we generalize their results to

higher dimensions.

1 Introduction

Let F

q

be a finite field with q elements and denote by F

q

((T

−1

)) the field of formal Laurent series. For

f ∈ F

q

((T

−1

)) let |f | = q

deg f

be the valuation induced by the generalized degree function. Set

L = {f ∈ F

q

((T

−1

)) : |f | < 1}.

Then, with the restriction of | · | to L, L is a compact topological group. Hence, there exists a (unique)

translation-invariant probability measure which will be denoted by m.

We are interested in the Diophantine approximation problem

f −

P

Q

<

1 q

n+ln

, deg Q = n, Q monic, (P, Q) = 1,

(1)

where f ∈ L, P, Q ∈ F

q

[T ] with Q 6= 0, and l

n

is a sequence of non-negative integers (subsequently,

we will use (·, ·) to denote the gcd, whereas h·, ·i will be used for pairs).

Concerning the number of solutions of (

1 ), Inoue and Nakada [

5 ] proved the following 0-1 law: the

number of solutions is either finite or infinite for almost all f ∈ L, the latter holding if and only if

∞

X

n=0

q

n−ln

_{= ∞.}

Moreover, the method of proof in [

5 ] also gives a quantitative result under one additional

assump-tion on l

n

: if l

n

≥ n, then the number of solutions of (

1 ) with deg Q ≤ N is given by

1 − q

−1

Ψ(N ) + O

Ψ(N )

1/2

(log Ψ(N ))

3/2+

,

where > 0 is an arbitrary small constant and Ψ(N ) :=

P

n≤N

q

n−ln

.

Key words:formal Laurent series, simultaneous Diophantine approximation, 0-1 law, strong law of large numbers. 2010 Mathematics Subject Classification: 11J61, 11J83, 11K60.

(22)

The purpose of this note is to prove generalizations of the above two results to multidimensional

Diophantine approximation. Therefore, consider

f

j

−

P

j

Q

<

1 q

n+l(j)n

, deg Q = n, Q monic, (P

j

, Q) = 1, j = 1, . . . , d,

(2)

where (f

1

, . . . , f

d

) ∈ L × · · · × L, P

j

, j = 1, . . . , d, Q ∈ F

q

[T ] with Q 6= 0, and l

(j)n

, j = 1, . . . , d are

sequences of non-negative integers. Moreover, set l

n

:=

P

dj=1

l

(j) n

.

Then, the first result above has the following extension to the multidimensional setting.

Theorem 1. The number of solutions of (

2 ) is either finite or infinite for almost all

(f

1

, . . . , f

d

) ∈

L × · · · × L, the latter holding if and only if

∞

X

n=0

q

n−ln

_{= ∞.}

₍₃₎

Moreover, also the second result admits an extension to higher dimensions.

Theorem 2. Assume that l

n

≥ n. Then, for almost all (f

1

, . . . , f

d

), the number of solutions of (

2 ) with

deg Q ≤ N is given by

c

0

Ψ(N ) + O

Ψ(N )

1/2+

,

where

> 0 is an arbitrary small constant, Ψ(N ) :=

P

n≤N

q

n−ln

, and

c

0

:=

X

Q1monic

· · ·

X

Qdmonic

µ(Q

1

)

|Q

1

|

· · ·

µ(Q

d

)

|Q

_d

|

1 |lcm(Q

1

, . . . , Q

d

)|

> 0

where

µ(·) is the Moebius µ function.

Remark

1. Observe that the error term in the above result for d = 1 is weaker than the corresponding

one in the result of Inoue and Nakada. The reason for this is that our method is completely different

from the approach used by the latter two authors (it is not obvious how to generalize their approach to

higher dimensions).

Notation.

We will use [D

1

, . . . , D

d

] to denote the lcm of the polynomials D

1

, . . . , D

d

. All sums will

be over monic polynomials. Logarithms in this paper just take on values ≥ 1, i.e. log

a

x should

be interpreted as max{log

_a

x, 1}. We will use both Landau’s notation f (x) = O(g(x)) as well as

Vinogradov’s notation f (x) g(x). Finally, will denote an arbitrary small positive number whose

value might change from one appearance to the next.

2 Proof of Theorem

1 First, note that the necessity of (

3 ) for the number of solutions of (

2 ) being infinite follows from a

standard application of the Borel-Cantelli lemma. Hence, we only have to focus on the sufficiency part.

For this purpose, we use a slight extension of the d-dimensional Duffin-Schaeffer theorem for formal

Laurent series due to Inoue [

4 ].

Theorem 3 (Inoue). Consider

f

j

−

P

j

Q

<

1 q

n+lQ(j)

, deg Q = n, Q monic, (P

j

, Q) = 1, j = 1, . . . , d,

(4)

(23)

where

(f

1

, . . . , f

d

) ∈ L×· · ·×L, P

j

, j = 1, . . . , d, Q with Q 6= 0, and l

(j)_Q

, j = 1, . . . , d are sequences

of non-negative integers. Assume that

X

Q

q

−l (1) Q −···−l (j) Q

_{= ∞}

and that for infinitely many

N

X

deg Q≤N

q

−l (1) Q −···−l (j) Q

_{< C}

X

deg Q≤N

q

−l (1) Q −···−l (j) Q

_ϕ(Q)

d

_/|Q|

d

_,

where

C is some positive constant. Then, (

4 ) has infinitely many solutions for almost all

(f

1

, . . . , f

d

) ∈

L × · · · × L.

Remark

2. Note that the result in [

4 ] is just stated for the special case l

_Q(1)

= . . . = l

_Q(d)

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