A NOTE ON DIOPHANTINE APPROXIMATION

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1 A NOTE ON DIOPHANTINE APPROXIMATION J. M. ALMIRA, N. DEL TORO, AND A. J. LÓPEZ-MORENO Received 15 July 2003 and in revised form 21 April 2004 We prove the existence of a dense subset of [0,4] such that for all α there exists a subgroup X α of infinite rank of Z[z]suchthatX α is a discrete subgroup of C[0,β]forall β α but it is not a discrete subgroup of C[0,β]foranyβ 0,α). Given a set of nonnegative real numbers Λ ={λ i } i=0,aλ-polynomial or Müntz polynomial) is a function of the form px) = n i=0 a i z λi n N). We denote by ΠΛ) the space of Λ-polynomials and by Π Z Λ):={px) = n i=0 a i z λi Πλ):a i Z for all i 0} the set of integral Λ-polynomials. Clearly, the sets Π Z Λ) are subgroups of infinite rank of Z[x] whenever Λ N,#Λ = by infinite rank, we mean that the real vector space spanned by X does not have finite dimension. In all what follows we are uniquely interested in groups of infinite rank). Now, it is well known that the problem of approximation of functions on intervals [a,b] by polynomials with integral coefficients is solvable only for intervals [a,b] of length smaller than four and functions f which are interpolable by polynomials of Z[x] on a certain set which we call the algebraic kernel of the interval [a,b]) a,b). Concretely, it is well known that Z[x]isadiscretesubgroupofC[a,b]wheneverb a 4 and 4 is the smallest number with this property for these and other interesting results about approximation by polynomials with integral coefficients, see [1, 3] and the references therein. See also the other references at the end of this note). This motivates the following concept. Definition 1. Given X a subgroup of infinite rank of Z[x], set α 0 X) = inf { α>0:x is a discrete subgroup of C[0,α] }. 1) In [1], Ferguson proved that α 0 X n ) = 4 1/n, where X n = Π Z nn) n N), and α 0 Π Z P)) = 1, where P denotes the set of prime numbers. Moreover, he observed that 1 α 0 Π Z Λ)) 4forallΛ N such that i=0 1/λ i ) =. We would like to observe that therelation1 α 0 Π Z Λ)) 4, holds for any infinite set Λ N. This clearly holds for the upper bound since Π Z Λ) Z[x]. To prove the lower bound it is enough to observe that x λn [0,α] = α λn 0whenn for all α 0,1), where Λ ={λ 0 <λ 1 < <λ n < } N. Copyright 2005 International Mathematics and Mathematical Sciences 2005:3 2005) DOI: /IJMMS

2 488 Note on Diophantine approximation In general, the numbers α 0 X) are not easy to compute. Moreover, they do not have nice properties in connection with the elementary operations of sets. For example, the knowledge of α 0 X i )i = 1,2) does not help for computing the numbers α 0 X 1 + X 2 )and α 0 X 1 X 2 ). Anyway, we have the following elementary properties. Proposition 2. Assume that X is a discrete subgroup of C[0,α]. Then X is also a discrete subgroup of C[0,β] for all β>α. Proof. By definition, X is a discrete subgroup of C[0,α] if and only if there exists a positive number ε>0 such that p [0,α] ε p X \{0}. 2) Hence, β α implies that X is a discrete subgroup of C[0,β], since p [0,β] p [0,α] ε p X \{0}. 3) Corollary 3. Assume that X 1 X 2 are two subgroups of Z[x] of infinite rank. Then α 0 X 1 ) α 0 X 2 ). Proof. We assume that α 0 X 2 ) <α 0 X 1 ) and take α α 0 X 2 ),α 0 X 1 )). Then X 2 is a discrete subgroup of C[0,α] since α>α 0 X 2 )), so that Hence p [0,α] ε p X 2 \{0}. 4) p [0,α] ε p X 1 \{0}, 5) since X 1 X 2. This implies that α α 0 X 1 ), a contradiction. Corollary 4. With the same notation as in the corollary above, it holds that α 0 X 1 + X 2 ) max{α 0 X 1 ),α 0 X 2 )} and α 0 X 1 X 2 ) min{α 0 X 1 ),α 0 X 2 )}. Moreover, there are examples where the inequalities are strict. Proof. We only prove that the inequalities are strict, in general since the other claim is a trivial consequence of the proposition above). To do this, we give the following examples: We set X 1 = Π Z 2N), X 2 = Π Z N \ 2N)andY 1 = Π Z 2N), Y 2 = Π Z 3N). Then Finally, Y 1 Y 2 = Π Z 6N), so that Z[x] = Π Z 2N)+Π Z N \ 2N), 6) ) 4 = α 0 Z[x] > 2 = α0 Π Z 2N) ) = α 0 Π Z N \ 2N) ). 7) α 0 Y1 Y 2 ) = α0 Π Z 6N) ) = 4 1/6 < 4 1/3 = min { α 0 Y1 ),α0 Y2 )}. 8) Now we prove the main result of this note.

3 J. M. Almira et al. 489 Theorem 5. Set = { α 0 X):X is a subgroup of infinite rank of Z[x] }. 9) Then R = [0,4]. Proof. Let α 0,4) be a transcendent number and we denote by J α = J0,α) the algebraic kernel of the interval [0,α]. We know that J α is a finite subset of [0,α] andα J α.thus ρα) = maxj α <α.foreachε 0,α ρα))/2), we take a function ϕ α,ε such that ϕ α,ε C 1 [0,α], ϕ α,ε x) = x for all x [0,ρα)], ϕ α,ε 0) = 0, ϕ α,ε α ε) = 4 ε, ϕ α,ε α) = 4+ε, and ϕ α,εx) > 0forallx [0,α]. This function satisfies ϕ k) α,ε) Jα = p k) J α, k = 0,1, for px) = x. Thus,ϕ α,ε belongs to the closure of Z[x] inc 1 [0,α] see [2, Theorem 12.1]), so that there exists a polynomial q α,ε Z[x]suchthat ϕ α,ε q α,ε C1 [0,α] < ε 2. 10) This implies that q α,ε is an strictly increasing function on [0,α], q α,ε 0) = 0 and there exists a certain number η α,ε α ε,α) suchthatq α,ε η α,ε ) = 4. Hence, q α,ε :[0,η α,ε ] [0,4] is an homeomorphism. We denote by Z[q α,ε ] the ring of polynomials with integral coefficients in the variable q α,ε x). Then Z[q α,ε ]isasubgroupofz[x] of infinite rank. We claim that α 0 Z [ qα,ε ]) = ηα,ε. 11) This is clear since, for all β 0,4] the map Φ β α,ε : C[0,β] C[0,qα,εβ)] 1 given by Φ β α,εu)= uq α,ε ) is an isometry of normed vector spaces which maps Z[x]ontoZ[q α,ε ] and the fact that α 0 Z[x]) = 4. If X is a subring of Z[x], then X is a discrete subset of C[0,α]ifandonlyif p [0,α] 1 for all p X \{0} since p [0,α] < 1 implies that lim n p n [0,α] = 0). This means that, for all ring X Z[x]wehavethat{α 0:X is a discrete subset of C[0,α]} is a closed set, so that α 0 X) = min { α>0:x is a discrete subgroup of C[0,α] }. 12) In particular, X is a discrete subset of C[0,α 0 X)]. Now, we note that the sets Z[q α,ε ]we used for the proof of the theorem above were subrings of Z[x], so that we have proved the existence of a dense subset of [0,4] such that for all α there exists a subgroup X α of infinite rank of Z[z]suchthatX α is a discrete subgroup of C[0,β]forallβ α but it is not a discrete subgroup of C[0,β]foranyβ 0,α). We end this note with two open problems: P1) is = [0,4]? P2) is ={α 0 Π Z Λ)) : Λ N,#Λ = }adense subset of [0,4]? In particular, is 4 an accumulation point of?

4 490 Note on Diophantine approximation Acknowledgment Research was partially supported by Junta de Andalucía, Grupo de Investigación Aproximación y Métodos Numéricos FQM References [1] L.B.O.Ferguson,Approximation by integral Müntz polynomials, Fourier Analysis and Approximation Theory, Vol. I Proc. Colloq., Budapest, 1976) G. Alexits and P. Turán, eds.), Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam, 1978, pp [2], Approximation by Polynomials with Integral Coefficients, Mathematical Surveys, vol. 17, American Mathematical Society, Rhode Island, [3] E. Hewitt and H. S. Zuckerman, Approximation by polynomials with integral coefficients, a reformulation of the Stone-Weierstrass theorem, Duke Math. J ), J. M. Almira: Departamento de Matemáticas, Universidad de Jaén, Linares Jaén), Spain address: jmalmira@ujaen.es N. Del Toro: Departamento de Matemáticas, Universidad de Jaén, Linares Jaén), Spain address: ndeltoro@ujaen.es A. J. López-Moreno: Departamento de Matemáticas, Universidad de Jaén, Linares Jaén), Spain address: ajlopez@ujaen.es

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