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ЧЕБЫШЕВСКИЙ СБОРНИК Том 11 Выпуск 1 (2010)
Труды VII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной памяти профессора Анатолия Алексеевича Карацубы
DIOPHANTINE APPROXIMATION ON THE CURVES WITH NONMONOTONIC ERROR FUNCTION IN THE pADIC CASE
It is shown that a normal (according to Mahler) curve in satisfies a convergent Khintchine Theorem with a non-monotonic error function.
Let Fn be the set of the normal functions
with n ^ 2, a = (a0,•••, an) £ Zn+^^d f : Zp ^ Zp, i = 1,... , n, be the normal functions. For F £ Fn define the height of F as H = H(F) = max0^n |aj |, Without loss of generality we will assume that f1(x) = ^^d an = H,
Let p be a prime number and Qp is the complete field of the p-adic numbers. Let ^,(U„) be the Lebesgue measure of a measurable set U, C R. Denote bv ^p the normalized Haar measure on Qp such that ^p(Zp) = 1, According to Mahler [1], the function f : Zp ^ Zp is called a normal if f has the form
where |a|p ^ 1, |an|p ^ 1 for all n and limn^„ |an|p = 0. Moreover, the normal function over Zp can be decomposed into the Tavlor series [2],
Let Pn be a set of polynomials P £ Z[x] of degree ^ n and ^ : N ^ R+ be a function. Given a polynomial P, H(P) will denote the height of P, i.e. the maximum of the absolute values of its coefficients. Let Wn(^) be a set of x £ R such that there are infinitely many P £ Pn satisfying
In 1924 Khintchine [3] proved that if the sum Y^h=i Ф(Н) converges, then ^(ЭД^(Ф)) is null, while if the sum diverges and Ф is decreasing, ^(R \ ЭД^(Ф)) is null. In 1969
Natalia Budarina
Аннотация
an fn(x) + ... + 02/2(2) + aifi(x) + a0
|P(x)| < Ф(Н(P)).
Sprindzuk proved that ^(Wn^)) = 0 if Ф(^ = H-w and w > n (see [4]), Baker [5] has improved Sprindzuk’s result and shown that ^(Wn^)) = 0 if Ф is monotonie function and
Ф(Н)” < to. (1)
H=1
In the same paper Baker conjectured that condition (1) can be replaced by the convergence of Ея=1 Hn-^(H), This conjecture was proved bv Bernik [6] in 1989, The divergence case was proved by Beresnevich [8] in 1999 who showed that ^(R \ ТС„(Ф)) = 0 if Ея=1 Hn-^(H) = то.
Ф
2005 Beresnevich [7] showed that the monotonicitv restriction can be avoid in the convergence case of Baker’s conjecture. The result of Beresnevich was generalized to non-degenerate curves in Rn [9].
Our main result below is a convergent Khintchine Theorem without monotonicitv condition for the normal curve in Z^,
Теорема 1. Let Ф : N ^ R+ be an arbitrary function (not necessarily 'monotonic) such that the sum Еь=1 h”^(h) converges. Let Ln^) be the set of x Є Zp such, that there are infinitely many F Є Fn satisfying
\F(x)\p < Ф(Я(F)). (2)
Then ^p(Ln(Ф)) = 0.
Since the sum Еь=1 h”^(h) converges then H”^(H) tends to 0 as H ^ то. Hence H^(H) = o(l) and Ф(^ = o(H-n).
The set Ln^) can be considered as the union of finite or countable number of the discs Ks, Further we fa one оf them K0, Without loss of generality we will assume that
radius(K0) < p-n. (3)
To prove the theorem two different cases concerning the size of \F/(x)\p are
considered. If x Є Ln^) then x must satisfy at least one of these cases infinitely
x
infinitely often has measure zero.
Case I. First the case of very small derivative is deal with.
x Є K0
\F(x)\p < Ф^), \F/(x)\p < H-1-v, v > 0, for infinitely many F Є Fn has measure zero.
This is proved using Theorem 1.5 from [10]. Using the notation in that theorem choose T0 = ... = Tn = H, Kp = H-1-v and 8 = H-n.
Proposition 1. Let p = min{n,v/(n +1)}. Let K0 c Qp and f = (f1;..., fn) : K0 ^ Qn be analytic non-degenerate map. For any x £ K0, one can find
a neighborhood V C K0 of x and A > 0 with the following property: if Bp C V any
ball then there exists E > 0 swcft that the set
UFeFn, 0<H(F{x £ Bp : |F(x)|p < H n, |F/(x)|p < H 1 v }
ftas measure at most EH- pA^p(Bp).
For a non-negative integer t and for any v > 0 we denote bv A(t) the set of x £ Bp
|F(x)|p << H-n, |F/(x)|p <H-1 -v (4)
holds for some F £ Fn with 2*-1 ^ H(F) < 2*. According to Proposition 1,
jjLp{A{t)) 2"^rr with w,A > 0, The set of i G K0 for which there are infinitely many F £ Fn satisfying (4) consists of points x £ Bp which belong to infinitely many sets A(t). The sum E,1 ^p(A(t)) converges for v > 0 and the Borel-Cantelli Lemma can be used to complete the proof of the lemma.
Case II. Now we consider the set Ln W of x £ K0 such that there are infinitely many F satisfy!ng |F(x)|p < ^(H^d |F/(x)|p > H-1-v.
Since the ring Zp of p-adic integers is compact then for F £ Fn we may define a point aF £ LnW such that
|F '(aF )|p = min |F /(x)|p.
xeLn(^)
Develop every function F £ Fn as a Tavlor series so that
,
F/(«f) + ^(m!)-1F(m)(«F)(x - )m-1 .
m=2 /
Using (3) and the fact that |F/(x)|p > H-1-v; we obtain
|F(x) - F(«F)|p = |F/(«F)|p|x - «F|p,
and from (2)
|x - |p ^ *(H)|F/(«f)|-1. (5)
Further the proof of theorem depends on the range for |F'(aF)|p.
Type 1. For F £ Fn, let a(F) be the set of x £ Ln(^) which satisfy
|F(x)|p < t(H), |F'(qf)|p > H"1/2. (6)
Given a F £ Fn define the disc
viiF) = {x eZp : \x- aF\p ^ {2p^ H\F\aF)\p)~1}. (7)
F(x) - F(aF) = (x - aF)
H
rtMF)) « H*(H)ft,MF)). (8)
Fix any function F for which a(F) = 0. Then we estimate the value ofp-adie norm of function F in any point x £ a1(F) by developing F as a Tavlor series:
Fix) = F(aF) + F'(aF)(x - aF) + V —F{m)(aF)(x - aF)m. (9)
m!
m=2
Bv (6), for n ^ 2 we have
|F(«f)|p < *(H) « H-2 S (2H)-1 (10)
H
|F'(aF)(x - aF)\p ^ \F\aF)\p(2p^H\F\aF)\p)-1 < (2H)~\ (11)
For m ^ 2 using the inequalities |m!|p S pm/(p-1) (see [11 ]), |F(m)(aF)|p S 1, (6)
and (7) we obtain
-^-F^m\aF)(x-aFy
m!
Using (9) - (11) and the previous inequality, we obtain
|F(x)|p < (2H)-1 (12)
for any x £ a1(F),
Fix the vector b = (H, an-1..^a^a^ andletthesubcl ass of Fn of functions with the same vector b be denoted by Fn(b), The number of different Fn(b) is « Hn-1, Let F1,F2 £ Fn(b), and assume that they have different coefficients a0. The number of different Fn(b) is « Hn-1, Let F1; F2 £ Fn(b), and assume that they have different coefficients a0, Assume that there is an x £ a1(F1) n a1(F2), Then by (12), |F1(x) -F2(x)|p < (2H)-1, On the other F1(x) - F2(x) is an integer not greater than 2H in absolute value. Therefore, |F1(x)-F2(x)|p ^ 1/|F1(x)-F2(x)|p > (2H)-1 that leads to a contradiction. Hence there is no such an x and a1(F1) n a1(F2) = 0, Hence,
J2 ^1(F) S MK0) (13)
F eFn(b)
Together with (8) this gives
p
£ Mp(^(F)) « H®(H)^p(A'0).
F eFn(b)
b
EE E M^(F)) « ^ Hn^(H)^p(K0) < to.
H=1 b FeF„(b) H=1
The Borel-Cantelli lemma can now be used to complete the proof.
Type 2. For F £ Fn, let a(F) be the set of x £ Ln W which satisfy
|F (x)|p < tf(H ),H-1-v < |F '(«f )|p <H-1/2, 0 <v< 1/4. (14)
Define the set a2(F) D a(F) as the set of points x which satisfy the inequality
\x — olf\p ^ H~2\F'(ap)\~1
for aF £ a(F), It is clear that
M*2(F)) = C1(p)H"2^(H), ^p(a(F)) = C2(p)H2^(H KMF)), (15)
where ci(p) > 0 (i = 1, 2) are the constants depen dent on p. Fix the vec tor b1 = (H, an-1..., a3, a2) and denote the subclass of Fn of functions with the same vector ^ by Fn(b1) The number of different sets Fn(b1^s « Hn-2,
Now for F £ Fn(b1) we estimate |F(x)|p where x £ a2(F), From definition of a2(F) it follows that |F'(aF)(x - aF)|p < H-2, For m ^ 2 bv (14) we have
I^F{n)(aF)(x - aF)n\v < (H~2H1+V)n ^ H~2+2v. n!
By Taylor’s formula and the previous estimates we get
|F'(x)|p « H-2+2v (16)
for any x £ a2(F) and v S 1/2.
Further we use essential and inessential domains introduced by Sprindzuk [4]. The domain a2(F) is called inessential if there is a function F £ Fn(b1) (with F = F) such that
Pl^W) > lh{a'22{F)),
and essential otherwise.
First, the inessential domains are investigated. Let the domain a2(F) is inessential and K1 = a2(F) P| ct2(F). Then
tbiKi) > \vPMF)) = C3{p)H-2\F'(aF)\-\ where c3(p) is a constant dependent on p.
Consider the new function R(x) = F(x) - F(x) = b1x + 60, where
F(x), F(x) £ Fn(b1) and max(|60|, |b1|) S 2H,
By (16),we obtain
|R(x)|p « H-2+2v (17)
for any x £ a2(F),
If we assume that 61 = 0 then |60|p « H-2+2v; but it is eontradieted to |60|p ^ |60|-1 » H-1, Hence, 61 = 0, By (17),we have
|x + 60/61 |p « H-2+2vI61I-1. (18)
Let K2 = {x £ K0 : the inequality (18) holds }, Then K1 C K^d ^p(K2) = c4(p)H-2+2v|61|-1, We have
H-2+2v|F'(«f)|-1 « ^p(K1) S ^p(K2) « H-2+2vI61I-1.
Hence, bv (14) and previous estimate we obtain |61|p « |F'(aF)|p « H-1/2, From (17) we obtain that |60|p « H-1/2, Suppose that s £ Z+ is defined by the inequalities ps S H < ps+1, For sufficiently large H we obtain that H1/2 x pts/2l. Hence 61 x p[s/2]611 and 60 x p[s/2]601 for 6n, 601 £ Z. We have 61x + 60 x pts/2l(611x + 601), where max |6111, |601| « H 1/2. Let R1(x) = 611x + 601, then H(R1) « H1/2, Then bv (17)
we obtain
\bnx+boi\p<^p[s/2]H-2+2v <^H~3/2+2v = H(Rl)~3+4v 5>0forv<^.
Bv Khintchine’s Theorem in Qp [4], we obtain that the set of x belonging to infinitely many dicks a2(F) has zero measure.
Now let a2(F) be essential domain. By the propertv of p-adie valuation every point x £ K0 belong to no more than one essential domain. Hence
M^2(F)) « ^p(K0).
F eFn(bi)
b1 « Hn-2
£ £ M^(F)) « Hn^(H)^p(K0).
bi FeFn(bi)
Finally, we obtain
Ma(F)) < to.
H=1 bi FeF„(bi)
many essential domains is of measure zero.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
[1] К. Mahler, Uber Transcendente p-adisce Zahlen, Composito Math. 2 (1935), 259-275.
[2] W.W. Adams, Transcendental numbers in the p-adie domain, Amer. J. Math. 88 2 (1966), 279-308.
[3] A. Khintehine, Einige Satze tiber Kettenbrtiehe mit Anwendungen auf die Theorie der Diophantisehen Approximationen, Math. Ann. 92 (1924), 115-125.
[4] V. Sprindzuk, Mahler’s problem in the Metric Theory of Numbers, Transl. Math. Monographs 25, Amer. Math. Soe,, Providence, R.I., 1969.
[5] A. Baker, On a theorem of Sprindzuk, Proc. Roy. Soc., London Ser. A 292 (1966), 92-104.
[6] V. I. Bernik, On the exact order of approximation of zero by values of integral polynomials, Acta Anth. 53 (1989), 17-28.
[7] V. V. Beresnevich, On a theorem of V. Bernik in the metric theory of
Diophantine approximation, Acta Arith. 117 1 (2005), 71-80.
[8] V. V. Beresnevich, On approximation of real numbers by real algebraic numbers, Acta Anth, 90 (1999), 97-112.
[9] N. Budarina, D. Dickinson, Diophantine approximation on non-degenerate curves with non-monotonic error function, Bulletin London Math. Soc. 41 1 (2009), 137-146.
[10] A. Mohammadi and A. Salehi Golsefidv, Simultaneous Diophantine appro-
p
Mathemtics).
[11] Burger, E. B. and Struppeck, Т., Does ~k\ R-eally Converge? Infinite Series
and p-adie Analysis, Amer. Math. Monthly 103 (1996), 565-577.
Получено 12.06.2010
