Malaysian Journal of Mathematical Sciences 9(2): 209-225 (205) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Non-classical Study on the Simultaneous Rational Approximation * Bellaouar Djamel and Boudaoud Abdelmadjid Laboratory of Pure and Applied Mathematics (LMPA), M sila University, B.P. 66, Ichbilia, 28000 M sila, Algeria E-mail: bellaouardj@yahoo.fr *Corresponding author ABSTRACT This study is placed in the framework of Internal Set Theory (Nelson, 977). Real numbers (ξ i ) i=,2,,k are called simultaneously approximable in the infinitesimal sense, if for every positive infinitesimal ε, there exist rational numbers ( p i q ) i=,2,,k such that { ξ i = p i q + ε i εq 0 ; i =,2,, k, where ( i ) i=,2,,k are limited numbers. Let (ξ 0, ξ,, ξ ω ) be a system of reals, with ω unlimited. In this paper, we will give a necessary condition for which (ξ i ) i=0,,,ω are simultaneously approximable in the infinitesimal sense. The converse of this condition is also discussed. Keywords: INTERNAL set theory, simultaneous rational approximation, infinitesimal sense.. INTRODUCTION. Elementary Nonstandard Notions We recall some definitions and facts from nonstandard analysis, real numbers, and sets that will be used in the proof of the main result. For more details, see Diener, 960; 995, Lutz and Goze, 98 and Nelson, 977.
Bellaouar Djamel & Boudaoud Abdelmadjid (a) A real number x is called unlimited if its absolute value x is larger than any standard integer n. So a nonstandard integer ω is also an unlimited real number. (b) A real number ε is called infinitesimal if its absolute value ε is smaller than for any standard n. n (c) A real number r is called limited if is not unlimited and appreciable if it is neither unlimited nor infinitesimal. (d) Two real numbers x and y are equivalent (written x y) if their difference x y is infinitesimal. (e) We distinguish two types of formulas: Formulas which do not contain the symbol "st" (for standard) are called internal, and formulas which do contain the symbol "st" are called external. (f) We call internal any set defined using an internal formula. (g) We call external any subset of an internal set defined using of an external formula for which a classical theorem at least is in default. Examples.. From Diener et Reeb, 960 in page 52, we have. Let ε be a positive real number (infinitesimal or not). The following sets are internal. [ ε, + ε], {x R ; εx }, and { n ; n N}. ε 2. Let ε be an infinitesimal positive real number. The set {x R ; x + ε x} is equal to R. i.e., it is internal. However, the set {x R ; x 0} () is external. In fact, if the set of () is internal, then it has the least upper bound a; which is neither infinitesimal nor appreciable (If a is infinitesimal, 2a and 3a are also. If a is appreciable, a is also). 2 That is, the Least Upper Bound Principle "A nonempty set of reals which is bounded above has the least upper bound" is in default. 20 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation 3. Let N σ be the set of limited (standard) positive integers, then N σ is external. In fact, if it is not we can apply the Principle of Mathematical Induction: Since is limited, then N σ. If s N σ, then s + N σ. Therefore N σ = N, which is impossible because there are unlimited positive integers. Lemma.2 (Robinson s Lemma). If (u n ) n 0 is a sequence such that u n 0 for all standard n, there exists an unlimited N such that u n 0 for all n N. Also, in this paper, we need to the following notions (see Cutland, 983, Diener, 995 and Van den Berg, 992). Definition.3. Let X be a standard set, and let (A x ) x X be an internal family of sets.. A union of the form G = stx X A x is called a pregalaxy; if it is external G is called a galaxy. 2. An intersection of the form H = stx X A x is called a prehalo; if it is external H is called a halo. Example.4. We have (a) N σ is a galaxy. (b) hal(0) = {x R ; x 0} is a halo. Theorem.5. No halo is a galaxy. Definition.6 (Shadow of a set). The shadow of a set A, denoted by A, is the unique standard set whose standard elements are precisely those whose halo intersects A. Theorem.7 (Cauchy s Principle). No external set is internal. For example, let ω be an unlimited positive integer. The shadow of (, 2,, ω, ω ) is equal to [0,]. Moreover, we see that ω ω ω ω en < ω for every Malaysian Journal of Mathematical Sciences 2
Bellaouar Djamel & Boudaoud Abdelmadjid standard positive integer n. From Cauchy s Principle there exists an unlimited integer n 0 which satisfies the previous inequality. In this work limited numbers are denoted by and infinitesimal numbers are denoted by ε or φ..2 Some Classical Results on the Simultaneous Rational Approximation We present some well known results on the simultaneous approximation of k p numbers ξ, ξ 2,, ξ k by fractions, p 2,, p ω. These results were q q q announced by Dirichlet s Theorem (Schmidt, 980 in page 27)), Kronecker s Theorem (Hardy and Wright, 960 in page 382), and many others. Theorem.8 (Dirichlet s Theorem). Let k be a positive integer, and let ξ, ξ 2,, ξ k be reals. For any integer Q >, we can find positive integers q, p, p 2,, p k such that q < Q k and qξ i p i ; for i =,2,, k. Q Theorem.9 (Hardy and Wright, 960, page 70). If ξ, ξ 2,, ξ k are any real numbers, then the system of inequalities ξ i p i q < q +μ, μ = k ; for i =,2,, k. has at least one solution. If one ξ i at least is irrational, then it has an infinity of solutions. Theorem.0 (Hardy and Wright, 960, page 70). Given ξ, ξ 2,, ξ k and any positive ε, we can find an integer q so that qξ i differs from an integer, for every i, by less than ε. Definition. (Schmidt, 962). A set of numbers ξ, ξ 2,, ξ r is linearly independent if no linear relation: a ξ + a 2 ξ 2 + + a r ξ r = 0, with integer coefficients, not all zero, holds between them. Theorem.2 (Kronecker s Theorem). If ξ, ξ 2,, ξ k are linearly independent, α, α 2,, α k are arbitrary, and Q and ε are positive, then there are integers q > Q, p, p 2,, p k 22 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation such that qξ i p i α i < ε, for i =,2,, k..3 How to give the infinitesimal sense to the simultaneous rational approximation? Let k be a positive integer and let (ξ, ξ 2,, ξ k ) be a system of reals. From Dirichlet s Theorem, for any integer Q >, the reals (ξ i ) i=,2,,k are simultaneously approximable by the rational numbers ( p i ) q i=,2,,k, with an error less than. That is, qq ξ i = p i q + e i, with e i qq and q < Qk ; i =,2,, k. For every positive infinitesimal ε, we can choose Q such that εq k 0, which implies εq 0. Thus, { ξ i = p i q + εγ i εq 0 with γ i ; i =,2,, k. εqq But, we have difficulty to prove that γ i is limited for i =,2,, k. Similarly, from Theorem.0, for every positive infinitesimal ε we have ξ i = p i + ε q i, with i = ; i =,2,, k. q Also, if ξ, ξ 2,, ξ k are linearly independent, we get the same result by using Theorem.2 whenever α = α 2 = = α k = 0. But we can not have the condition εq 0. The following definition gives a new sense to the simultaneous rational approximation of k numbers. Definition.3. Let (ξ, ξ 2,, ξ k ) be a system of reals, with k. The reals (ξ i ) i=,2,,k are said to be simultaneously approximable in the infinitesimal sense, if for every positive infinitesimal ε, there exist rational numbers ( p i ) such that q i=,2,,k { ξ i = p i q + ε i εq 0, ; i k, (2) Malaysian Journal of Mathematical Sciences 23
where ( i ) i=,2,,k are limited numbers. Bellaouar Djamel & Boudaoud Abdelmadjid The real ε allows to control the error ε i and the denominator q. In fact, from classical results of the simultaneous approximation, ξ i = p i q + e i for i =,2,, k. Then, for every positive infinitesimal ε, ξ i = p i q + ε i, with i = e i ε. In Definition.3, we have added two conditions: i is a limited for i =,2,, k and εq 0. Notation.4. Let S A ( 0) denote the set of all systems (ξ, ξ 2,, ξ k ), with k for which (ξ i ) i=,2,,k satisfy (2). In this paper, we will prove that S A ( 0) is a non-empty set. Also, for a given system of reals (ξ 0, ξ,, ξ ω ), with ω + we ask if there is a necessary and a sufficient condition on the reals (ξ i ) i=0,,,ω for which (ξ 0, ξ,, ξ ω ) S A ( 0). We are in a position to give our main results. lemma. 2. MAIN RESULTS To prove that S A ( 0) is a non-empty set, we need the following Lemma 2.. Let N, ω be two unlimited positive integers. Let ε be an infinitesimal positive real number. If εn ω is not infinitesimal, then there exists an integer i 0 {, 2,, ω } such that εn ω i 0 0 and εn ω (i 0+) 0. Proof. Let α be an appreciable number strictly less than εn ω which we may, because εn ω 0. Since 0 εn ω ω < εn < εn 2 < < εn ω 2 < εn ω 0, there exists an integer s {, 2,, ω } such that There are two cases to consider. εn ω (s+) α < εn ω s. εn ω (s+) 0, Lemma 2. is proved by taking i 0 = s. εn ω (s+) 0. Since N +, we have 24 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation εn ω (s+2) = εnω (s+) N α N 0. Also, Lemma 2. is proved by taking i 0 = s +. Theorem 2.2. S A ( 0) is a non-empty set. Proof. In the following proposition, we give a system containing an unlimited number of reals that satisfies (2). That is, we prove that S A ( 0) contains many systems of the form ( ξ 0, ξ,, ξ k ), with k is an unlimited.. Proposition 2.3. Let N, ω be two unlimited positive integers. Then, ( N ω, N ω,, N, ) S A( 0). (3) Proof. Let ε be an infinitesimal positive real number, there are two cases. A) εn ω 0. For every i = 0,,, ω, we have In this case, Proposition 2.3 is proved. Ni = { Nω i N ω + ε. 0 = p i q + ε i εn ω = εq 0. B) εn ω 0. Here we distinguish two cases. B.) εn ω = a with a is an appreciable. In this case, we can write the system of (3) as follows: 0 N ω N ω + ε p 0 a q + ε 0 p + ε. 0 N ω Nω q + ε = N = p 2 + ε. 0 N ω 2 Nω q + ε, 2 N ( ) ω p ω ( N ω + ε. 0 ) ( q + ε ω ) Malaysian Journal of Mathematical Sciences 25
Bellaouar Djamel & Boudaoud Abdelmadjid where εq = εn ω = εnω this case. N = a N 0. So, Proposition 2.3 is proved for B.2) εn ω +. In this case we also distinguish two cases. B.2.) The real εn ω is infinitesimal. Since ω 0, it follows that 0 p 0 + ε N ω Nω εn ω q + ε 0 p + ε. 0 N ω Nω q + ε = N = p 2 + ε. 0 N ω 2 Nω q + ε, 2 N ( ) ω p ω ( N ω + ε. 0 ) ( q + ε ω ) where εq = εn ω 0. Proposition 2.3 is proved. B.2.2) The real εn ω is not infinitesimal. Let i 0 {, 2,, ω } be the integer constructed in Lemma 2., then 0 N ω (i 0+) + ε p 0 εn ω q + ε 0 N ω p q + ε N ω p i0 N ω i 0 N ω (i 0+) N ω (i 0+2) N ( ) = ( with εq = εn ω (i 0+) 0. 0 N ω (i 0+) + ε εn ω 0 N ω (i 0+) + ε εn ω i 0 N ω (i 0+) + ε. 0 N N ω (i 0+) + ε. 0 N ω (i 0+2) N ω (i 0+) + ε. 0 N ω (i 0+) N ω (i 0+) + ε. 0 ) This completes the proof of Proposition 2.3. εn = ( q + ε i 0 p i0 + q + ε i 0 + p i0 +2 q + ε i 0 +2 p ω q + ε ω p ω q + ε ω ) 26 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation Lemma 2.4. Let ω be an unlimited positive integer, and let ( ξ 0, ξ,, ξ ω ) be a system of reals satisfying the following properties: (a) ξ 0 ξ ξ ω (b) ξ i+ ξ i = d i > 0 for i = 0,,, ω d (c) i = a d i for i =, 2,, ω. i Then, (ξ 0, ξ,, ξ ω ) S A ( 0). Proof. Assume, by way of contradiction, that the reals (ξ i ) i=0,,,ω are simultaneously approximable in the infinitesimal sense. In particular, for ε = d 0 0 we have { ξ i = p i q + ε i εq 0, (4) where p i q is a rational and i is a limited for every i = 0,,, ω. Let i 0 be an unlimited positive integer strictly less than ω and satisfying i 0 < Nεq (5) for a given limited integer N > 2 (which we may, because = + ). εq d 0 q Since the reals (a i ) i=,2,,ω are all appreciable then, for any standard integer n n, the number S n = i= a a 2 a i is also an appreciable. Next, consider the set {n {, 2,, ω } ; + a a 2 a i < i 0 + }, n i= (6) which is internal and contains N σ. According to the Cauchy s Principle there exists an unlimited integer n 0 that satisfies (6). On the other hand, since Malaysian Journal of Mathematical Sciences 27
Bellaouar Djamel & Boudaoud Abdelmadjid n 0 ξ n0 ξ 0 = d 0 + d +... +d n0 = ε d i, i=0 d 0 (7) and d i d i = a i, for i =, 2,, ω. From (4) and (7), we have n 0 ξ n0 ξ 0 = ε ( + a a 2 a i ) = p n 0 p 0 q i= + ε. (8) We use the fact that n 0 satisfies (6). Then from (5), (6), and (8) we get (p n0 p 0 ) + εq < N. Since εq 0, it follows that p n0 p 0 < 2 N. Now we prove that p n0 > p 0. First, it suffices to prove that the n 0 number + i= a a 2 a i is unlimited. In fact, consider the following set {m N ; m n 0 and + a a 2 a i > m}, which is internal and contains N σ, because for all limited integers s we have s m i= + a a 2 a i = + s + φ s > s, i= where φ s 0 (positive or negative). From Cauchy s principle there exists an unlimited integer m 0 (with m 0 n 0 ) such that n 0 + a a 2 a i + a a 2 a i > m 0 +. i= m 0 i= 28 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation We assume that p n0 = p 0, by (8) we get n 0 + + a a 2 a i =, i= which is a contradiction. Therefore p n0 p 0. Moreover, if p n0 < p 0, by using (8) again, we obtain > p 0 p n0 εq +, because ξ n0 > ξ 0. Which is a contradiction, since is limited. Recall that p n0 and p 0 are positive integers, and since p n0 > p 0 it follows that 2 N >. Which leads to a contradiction with the hypothesis of N > 2. This completes the proof. Theorem 2.5. Let ω be an unlimited positive integer, and let ( ξ 0, ξ,, ξ ω ) be a system of reals. If ( ξ 0, ξ,, ξ ω ) contains a standard interval [a, b] with a < b then the reals (ξ i ) i=0,,,ω are not simultaneously approximable in the infinitesimal sense. That is, we will prove the necessary condition given by: ( ξ 0, ξ,, ξ ω ) S A ( 0) a, b N σ : [a, b] ( ξ 0, ξ,, ξ ω ) (N) Proof. Since ( ξ 0, ξ,, ξ ω ) contains a standard interval [a, b] with a < b, there exists a subsystem ( ξ i0, ξ i,, ξ ik ) ( ξ 0, ξ,, ξ ω ) such that { ( ξ i 0, ξ i,, ξ ik ) = [a, b] ξ i0 < ξ i < < ξ ik, where k +. We prove that a ξ i0 ξ i ξ ik b. In fact, suppose the contrary, i.e., there exists m {,2,..., k} such that Since ξ im ξ im, it follows that ξ im ξ im. Malaysian Journal of Mathematical Sciences 29
Bellaouar Djamel & Boudaoud Abdelmadjid ξ im + ξ im 2 ξ i m + ξ im 2 [a, b]. Which is a contradiction because the number ξ im +ξ im 2 the system ( ξ i0, ξ i,, ξ ik ). does not belong to Put d s = ξ is+ ξ is for s = 0,,, k, then max 0 s k (d s ) 0. Let γ be an unlimited real number such that and this by using Robinson s Lemma. λ = γ max (d s) 0, 0 s k Now, we choose a system of N elements (θ r ) r=0,,,n among the numbers (ξ is ) s=0,,,k as the following way θ 0 = ξ i0, and θ r = ξ im r is the nearest element strictly less than θ 0 + r λ; r =,2,, N. where m r {0,,, k} and N is an unlimited integer, with Nλ 0. Then, we prove the conditions (a), (b) and (c) of the Lemma 2.4 for the new system ( θ 0, θ,, θ N ). In fact, from the construction of (θ r ) r=0,,,n we see that θ 0 θ θ N ξ i0 and θ r+ θ r = D r > 0 ; 0 r N. Thus, (a) and (b) are satisfied. For the proof of (c), we put δ r = ξ i0 + r λ θ r ; r = 0,,..., N. Then, δ r ξ ξ im r+ i = d m r i, because ξ m r i m is the nearest element strictly r less than θ 0 + r λ. Moreover, we have δ r λ = δ r γ max (d s) 0 s k γ ( δ r ) d im γ 0. r Therefore, for every r = 0,,..., N, there exists an infinitesimal real number φ r such that δ r = λ φ r. Hence 220 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation θ r+ θ r = λ δ r+ + δ r = λ λ φ r+ + λ φ r ; for r = 0,,, N. It follows for every r {, 2,, N } that D r D r = θ r+ θ r θ r θ r = φ r+ + φ r φ r + φ r. Using Lemma 2.4 we can also conclude that ( θ 0, θ,, θ N ) S A ( 0) and therefore (ξ 0, ξ,, ξ ω ) S A ( 0). This completes the proof of Theorem 2.5. Corollary 2.6. The set S A ( 0) does not contain countable systems. Proof. Let (ξ 0, ξ,, ξ ω,... ) be a countable system of reals. There exists a subsystem ( ξ i0, ξ i,, ξ ik ) satisfying the conditions of Lemma 2.4, with k. Hence ( ξ i0, ξ i,, ξ ik ) S A ( 0), and therefore (ξ 0, ξ,, ξ ω,... ) S A ( 0). In the following result, for a real number x, let {x} and [x] denote the fractional part and the integer part of x, respectively. Corollary 2.7. Let ω be an unlimited positive integer. If (ξ 0, ξ,, ξ ω ) S A ( 0) then, for every limited integer c, ({cξ 0 }, {cξ },, {cξ k }) does not contain any standard interval [a, b], with a < b. Proof. Suppose that there exists a subset of positive integers: ( i 0, i,, i k ) (0,,, ω), with k +, and there is a limited integer c 0 such that ({c 0 ξ i0 }, {c 0 ξ i },, {c 0 ξ ik }) = [a, b] where a and b are standard real numbers (a < b ), and we prove that (ξ 0, ξ,, ξ ω ) S A ( 0). In fact, from Theorem 2.5, we get ({c 0 ξ i0 }, {c 0 ξ i },, {c 0 ξ ik }) S A ( 0). (9) It suffices to show that (c 0 ξ i0, c 0 ξ i,, c 0 ξ ik ) S A ( 0). Suppose the contrary. Then for every positive infinitesimal ε there exist rational numbers ( P is ) such that Q s=0,,,k Malaysian Journal of Mathematical Sciences 22
Bellaouar Djamel & Boudaoud Abdelmadjid { {c 0ξ is } = P i s [c 0 ξ is ]Q + ε Q εq 0 ; 0 s k, because {c 0 ξ is } = c 0 ξ is [c 0 ξ is ], for s = 0,,, k. Thus, ({c 0 ξ i0 }, {c 0 ξ i },, {c 0 ξ ik }) S A ( 0). Which contradicts the expression (9). Finally, since c 0 is a limited integer, we have (ξ i0, ξ i,, ξ ik ) S A ( 0), and therefore (ξ 0, ξ,, ξ ω ) S A ( 0). This completes the proof. 3. REMARKS AND EXAMPLES In this section, we give certain remarks and examples about the necessary condition stated in Theorem 2.5. Remark 3.. The converse of (N) is false. In the following corollary, we give a system of real numbers ( ξ 0, ξ,, ξ ω ) with ω +, whose elements are not simultaneously approximable in the infinitesimal sense but its shadow is different from a standard interval [a, b]. Corollary 3.2 (Counterexample). Let f be the exponential function. For every unlimited positive integer ω, we have ( f(0), f(),, f(ω) ) S A( 0). (0) Proof. Suppose that the reals of (0) are simultaneously approximable in the infinitesimal sense. Then, for ε = 0, there exist rational numbers f(ω) ( p i ) q i=0,,,ω such that 222 Malaysian Journal of Mathematical Sciences
Non-classical Study on the Simultaneous Rational Approximation { f(i) = p i q + ε i εq 0, where i is a limited number for every i = 0,,, ω. Using Cauchy s principle, there exists an unlimited positive integer i 0 such that f(i 0 ) < γ 2, () where γ = εq +. Since f is increasing, we have i 0 < ω. In fact, if i 0 ω it follows that f(ω) < f(ω). Which is impossible. 2q Now, we put s 0 = ω i 0. From the hypothesis, there exist p s0 q, p ω q such that f(s 0 ) f(ω) = ε(f(i 0) ) = p s 0 p ω + ε. (2) q Using () and (2), we get (p s0 p ω ) + εq < 2. (3) It follows from (2) that p s0 p ω because f(i 0 ) +. Moreover, if p s0 < p ω then > p ω p s0 εq +. which we may, because >. A contradiction, since is a limited. f(s 0 ) f(ω) Thus, p s0 > p ω. Finally, from (3) we have p s0 p ω < 2, since εq 3 0. Which is impossible. Remark 3.3. Let f be the function of Corollary 3.2, we put A = (,,, ). Since f is standard, then A = A is not an interval. f(0) f() f(ω) Malaysian Journal of Mathematical Sciences 223
Bellaouar Djamel & Boudaoud Abdelmadjid Corollary 3.4 (An example of Theorem 2.5). Let ω be an unlimited positive integer. Then, ( ω, 2 ω,, ω ω, ω ω ) S A( 0). Proof. It is clear that ( ω, 2 ω,, ω ω, ω ω ) = [0,]. Thus we get the result by using Theorem 2.5. Moreover, for any standard interval [a, b], with a < b, there exists a system of reals (ξ 0, ξ,, ξ ω ), with ω + such that a ξ 0 ξ ξ ω b, and also from Theorem 2.5, (ξ 0, ξ,, ξ ω ) S A ( 0). Remark 3.5. Let (ξ 0, ξ,, ξ k ) be an arbitrary system of real numbers. From the proof of Corollary 2.7, it is clear that (ξ 0, ξ,, ξ k ) S A ( 0), if and only if ({ξ 0 }, {ξ },, {ξ k }) S A ( 0), where {x} represent the fractional part of x. We can therefore deduce that if (ξ 0, ξ,, ξ k ) S A ( 0) then (ξ 0, ξ,, ξ k, {ξ 0 }, {ξ },, {ξ k }) S A ( 0). 4. CONCLUSION In this paper, we aim to give a necessary condition for a system of real numbers (ξ 0, ξ,, ξ ω ) to be in S A ( 0), where ω is an unlimited positive integer. However, a sufficient condition remains an open problem. ACKNOWLEDGEMENTS The first author would like to express his hearty thanks to the referee for his thorough scrutinizing the paper and for many useful comments which improved the earlier version completely, resulting in this improver version. 224 Malaysian Journal of Mathematical Sciences
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