Diophantine Approximation Generalized - Springer Link

c Pleiades Publishing, Ltd., 2012. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2012, Vol. 276, pp. 193–207.

Diophantine Approximation Generalized Ladislav Miˇ s´ık a and Oto Strauch b Received August 2011

Dedicated to the memory of Professor Anatolii Alekseevich Karatsuba Abstract—In this paper we study the set of x ∈ [0, 1] for which the inequality |x − xn | < zn holds for infinitely many n = 1, 2, . . . . Here xn ∈ [0, 1) and zn > 0, zn → 0, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution functions of sequences, we find the asymptotic density of n for which |x − xn | < zn , where x is a discontinuity point of some distribution function of xn . Generally, we also prove, for an arbitrary sequence xn , that there exists zn such that the density of n = 1, 2, . . . , xn → x, is the same as the density of n = 1, 2, . . . , |x − xn | < zn , for x ∈ [0, 1]. Finally we prove, using the longest gap dn in the finite sequence x1 , x2 , . . . , xn , that if dn ≤ zn for all n, zn → 0, and zn is non-increasing, then |x − xn | < zn holds for infinitely many n and for almost all x ∈ [0, 1]. DOI: 10.1134/S0081543812010166

1. INTRODUCTION Some authors extend the following problem of diophantine approximation: x − p < f (q), p, q are integers, gcd(p, q) = 1, q > 0, q

(1)

for infinitely many p, q, to the form |x − xn | < zn ,

(2)

where x ∈ [0, 1], xn ∈ [0, 1), zn ≥ 0, zn → 0, n = 1, 2, . . . , and study the set ∞ D (xn )∞ n=1 , (zn )n=1 = x ∈ [0, 1]; |x − xn | < zn for infinitely many n . Recently this viewpoint was presented by D. Berend and A. Dubickas [1], and they found, among ∞ )∞ other things, the Hausdorff dimension of D((xn n=1 , (zn )n=1 ). However, (2) was first used by ∞ J. Lesca [13] to study the sequences xn for which n=1 zn = ∞ implies that the Lebesgue measure ∞ 2 ∞ of D((xn )∞ n=1 , (zn )n=1 ) is 1. In [22] the L discrepancy theory is applied to such sequences (xn )n=1 . In [20] Strauch studied the inequality zn (3) |x − xn | < z in connection with Dini derivates of the function f : [0, 1] → R, where f (xn ) = zn for all n and f (x) = 0 otherwise.

(4)

The study of (1) was inspired by the effort to replace f (q) = 1/q 2 by a smaller function f (q) for which every irrational x can be approximated by infinitely many fractions p/q such that (1) holds. a Department of Mathematics, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic. b Mathematical Institute, Slovak Academy of Sciences, Stef´ ˇ anikova 49, SK-814 73 Bratislava, Slovakia.

E-mail addresses: [email protected] (L. Miˇs´ık), [email protected] (O. Strauch).

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ˇÍK, O. STRAUCH L. MIS

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This question answered by A. Hurwitz in 1891, who showed that the best possible function is √ was 2 information for this f (q) was f (q) = 1/( 5q ). The application of Lebesgue measure to bring more found by A. Khintchine [10] in 1924: If q 2 f (q) is non-increasing and ∞ q=1 qf (q) diverges, then (1) has infinitely many integer solutions for almost all x. Originally, he did not assume gcd(p, q) = 1. This leads to the Duffin–Schaeffer conjecture: Let f (q) be a function defined on the positive integers and let ϕ(q) be the Euler totient function. The Duffin and Schaeffer conjecture (D.S.C.) says that for an arbitrary function f (q) ≥ 0 defined on positive integers (zero values are also allowed for f (q)) the diophantine inequality (1) has infinitely many integer solutions ∞ p and q for almost all x ∈ [0, 1] (in the sense of Lebesgue measure) if and only if the series q=1 ϕ(q)f (q) diverges. The basic general reference books on the D.S.C. are V.G. Sprindzuk [18] and G. Harman [8]. A plan of the paper is the following: In Section 2 we give an overview of some types of sequences used in (2). In Section 3 we briefly summarize basic results on (1) and (2) up to now. In Section 4 by using uniform distribution theory we study an asymptotic density of n for which |x − xn | < zn , probably a new direction. Finally, we find conditions which imply that the Lebesgue measure of ∞ D((xn )∞ n=1 , (zn )n=1 ) is 1. 2. NOTATION AND DEFINITIONS In connection with (2) there are the following special types of sequences (xn )∞ n=1 . ∞ ∞ • A sequence ∞ (xn )n=1 is said to be eutaxic if for every non-increasing sequence (zn )n=1 the divergence of n=1 zn implies that

lim #{n ≤ N ; |x − xn | < zn } = ∞

N →∞

holds for almost all x ∈ [0, 1]. • If furthermore

#{n ≤ N ; |x − xn | < zn } = 1, N →∞ 2 N n=1 zn lim

then (xn )∞ n=1 is called strongly eutaxic. Let X = ∞ m=1 Im be a decomposition of an open set X ⊂ [0, 1] into a sequence Im , m = 1, 2, . . . , of pairwise disjoint open subintervals of [0, 1] (empty intervals are allowed). • An infinite sequence (xn )∞ n=1 in [0, 1) is said to be quick if for every open set X ⊂ [0, 1] with the Lebesgue measure |X| < 1 covering the sequence xn (i.e., xn ∈ X, n = 1, 2, . . . ), there exists a constant c(X) > 0 such that #{m ∈ N; ∃n ≤ N such that xn ∈ Im } ≥ c(X) > 0 N

for N = 1, 2, . . . .

Here |X| denotes the Lebesgue measure of X. • A sequence (xn )∞ n=1 is said to be uniformly quick (abbreviated u.q.) if for any open set X ⊂ [0, 1] which covers (xn )∞ n=1 we have #{m ∈ N; ∃n ≤ N such that xn ∈ Im } = 1 − |X|. N →∞ N lim

• If the above limit holds for a special sequence of indices N1 < N2 < . . . , then (xn )∞ n=1 is said to be almost u.q. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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From uniform distribution theory we employ the following (see [12] and [24]): • A sequence (xn )∞ n=1 can have infinitely many distribution functions (d.f.s) defined as all possible limits #{n ≤ Nk ; xn ∈ [0, x)} → g(x) FNk (x) = Nk as k → ∞. We shall denote the set of all such g(x) by G((xn )∞ n=1 ). Some special d.f.s are cα (x), a one-jump d.f. having a jump of height 1 at x = α; hα (x), a constant d.f. with hα (x) = α for x ∈ (0, 1). ∞ • A sequence (xn )∞ n=1 is called a maldistributed sequence in [0, 1) if the set G((xn )n=1 ) contains all one-step d.f.s cα (x).

• [x] denotes the integer part and {x} the fractional part of x. • Let nk , k = 1, 2, . . . , be an increasing sequence of positive integers. The lower asymptotic den∞ ∞ sity d((nk )∞ k=1 ) and the upper asymptotic density d((nk )k=1 ) of the sequence (nk )k=1 are defined by d((nk )∞ k=1 ) = lim inf x→∞

#{k ∈ N; nk ∈ [0, x]} k = lim inf , k→∞ nk x

d((nk )∞ k=1 ) = lim sup x→∞

#{k ∈ N; nk ∈ [0, x]} k = lim sup . x k→∞ nk

∞ ∞ If d((nk )∞ k=1 ) = d((nk )k=1 ), we say that the sequence (nk )k=1 possesses the asymptotic density d((nk )∞ k=1 ), given by this common value.

• Extremal discrepancy of a finite sequence x1 , x2 , . . . , xN in [0, 1) is defined as #{n ≤ N ; xn ∈ I} N − |I|, DN = D((xn )n=1 ) = sup N I⊂[0,1] where I is an interval with Lebesgue measure |I|. An infinite sequence xn , n = 1, 2, . . . , is uniformly distributed (u.d.) if D((xn )N n=1 ) → 0 as N → ∞. 3. KNOWN RESULTS I. Eutaxic sequences were introduced by J. Lesca [13]. (i) He proved that if θ is irrational, then the sequence of fractional parts ({nθ})∞ n=1 is eutaxic if and only if θ has bounded partial quotients. (ii) M. Reversat proved [17] the same for the strong eutaxy of ({nθ})∞ n=1 . (iii) Applying L2 discrepancy theory, Strauch proved in [22] that N 1 (xm − zm , xm + zm ) ∩ (xn − zn , xn + zn ) = 1 lim N 2 N →∞ 2 m,n=1 n=1 zn

implies the existence of a sequence N1 < N2 < . . . for which #{n ≤ Nk ; |x − xn | < zn } =1 k k→∞ 2 N n=1 zn lim

for almost all x ∈ [0, 1].

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II. Quick and u.q. sequences were introduced and studied by O. Strauch [19] in connection with the D.S.C., and he proved the following: ∞ ∞ (i) Any ∞quick sequence (xn )n=1 is eutaxic; i.e., for every non-increasing sequence (zn )n=1 , zn → 0, n=1 zn = ∞, for almost all x ∈ [0, 1], the inequality |x − xn | < zn holds for infinitely many n. It is conjectured that the eutaxic and quick sequences coincide.

(ii) Any u.q. sequence (xn )∞ n=1 is u.d. in (0, 1] and it is also strongly eutaxic. (iii) The sequence xn = nθ mod 1, n = 1, 2, . . . , is u.q. if and only if the simple continued fraction expansion of the irrational θ has bounded partial quotients (cf. O. Strauch [21, Theorem 3]). III. D. Berend and A. Dubickas [1] proved the following: ∞ (i) Let (xn )∞ n=1 be an arbitrary dense sequence in [0, 1], and (zn )n=1 be an arbitrary sequence ∞ of positive numbers. Then the set D((xn )∞ n=1 , (zn )n=1 ) is an uncountable dense subset of the interval [0, 1]. (ii) Let (xn )∞ an arbitrary sequence in [0, 1], and (zn )∞ n=1 be n=1 be an arbitrary sequence of ∞ s ∞ positive numbers. If n=1 zn < ∞ for some 0 < s < 1, then dimH D((xn )∞ n=1 , (zn )n=1 ) ≤ s (the Hausdorff dimension). (iii) For any sequence zn > 0, n = 1, 2, . . . , limn→∞ zn = 0, there exists a well-distributed ∞ ∞ (xn )∞ n=1 in [0, 1) such that dimH D((xn )n=1 , (zn )n=1 ) = 0. ∞ ∞ (iv) For every zn , n = 1, 2, . . . , n=1 zn = ∞, there exists (xn )n=1 in [0, 1] such that ∞ ∞ D((xn )n=1 , (zn )n=1 ) = [0, 1]. (v) For the sequence xn of all dyadic rational numbers from [0, 1) ordered as 0/2, 1/2, 1/4, 3/4, ∞ 1+ε , then 1/8, 3/8, 5/8, 7/8, 1/16, . . . and zn = 1/n we have D((xn )∞ n=1 , (zn )n=1 ) = [0, 1]. If zn = 1/n ∞ ∞ D((xn )n=1 , (zn )n=1 ) is uncountable dense with zero Lebesgue measure and containing no algebraic numbers. Additional information (cf. [19]): For an arbitrary non-increasing sequence zn , zn > 0, ∞ ∞ ∞ z n = 1, 2, . . . , zn → 0, if n=1 n = ∞, then |D((xn )n=1 , (zn )n=1 )| = 1, i.e., xn is an eutaxic sequence. 2/3, 1/4, 3/4, (vi) Consider the sequence xn of Farey fractions from (0, 1] ordered as 1/1, 1/2, √ 1/3, −1 + ε)/√n, then = ( 3π 1/5, 2/5, 3/5, 4/5, 1/6, . . . and let ε be an arbitrary positive number. If z n √ ∞ 2 5)−1 + ε)/n, then D((x )∞ , (z )∞ ) = [0, 1] \ Q. D((xn )∞ n n=1 n n=1 n=1 , (zn )n=1 ) = [0, 1]. If zn = (3(π Additional information (cf. [19]): xn is an eutaxic sequence, again. Additional notes: (v) By definition of eutaxic sequences, for xn = nθ mod 1, if θ has bounded partial quotients, ∞ ∞ then ∞ the measure |D((xn )n=1 , (zn )n=1 )| is 1, for an arbitrary monotone sequence zn > 0, zn → 0, n=1 zn = ∞. ∞ ∞ (vi) In all cases, if ∞ n=1 zn < ∞, then |D((xn )n=1 , (zn )n=1 )| = 0. IV. The D.S.C. is one of the most important unsolved problems in metric number theory until now (cf. also G. Harman [8, p. 53]). ∞(i) By the Borel–Cantelli lemma, (1) has only finitely many solutions for almost all x if q=1 ϕ(q)f (q) converges. (ii) By the Gallagher ergodic theorem [6] the set of all x ∈ [0, 1] for which (1) has infinitely many integer solutions has measure either 0 or 1. (iii) A. Haynes, A. Pollington and S. Velani [9] proved the following: If ∞ q=1 ϕ(q)f (q) = ∞, then the set of x for which (1) has infinitely many integer solutions p and q is of Hausdorff dimension 1. (iv) We say that a set X of sequences qn , n = 1, 2, . . . , of distinct positive integers and a set Y of functions f (x) satisfy the D.S.C. if for any (qn )∞ n=1 ∈ X and f (x) ∈ Y the divergence of PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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∞

implies that for almost all x ∈ [0, 1] there exist infinitely many n such that the diophantine inequality x − p < f (qn ), gcd(p, qn ) = 1, (5) qn n=1 ϕ(qn )f (qn )

has an integer solution p. (v) The inequality |x − xn | < zn covers (5) in the form with (xn )∞ n=1 composed of a block sequence (An )∞ , n=1

aϕ(qn ) 1 a2 , ,..., , gcd(ai , qn ) = 1, An = qn qn qn and with zi = f (qn ) for the terms xi ∈ An . (vi) There are three types of results on qn and f (x) satisfying the D.S.C.: 2 (a) any one-to-one sequence (qn )∞ n=1 and a special f (x), for example, f (x) = c/x (P. Erd˝os [4]);

(b) any f (x) ≥ 0 and a special (qn )∞ n=1 , for example, A.C. Schaeffer [3]);

ϕ(qn ) qn

≥ c > 0 (R.J. Duffin and

(c) special qn and f (x), for example, f (qn )qn > c1 (ϕ(qn )/qn )c2 for some c1 , c2 > 0 (G. Harman [8]). V. Applying the theory of real functions to the Riemann type function defined in (4), Strauch in [20] proved the following: (i) If xn is dense in [0, 1] and zn → 0, the sets X1 = x ∈ [0, 1]; for every z > 0, (3) and xn > x hold for infinitely many X2 = x ∈ [0, 1]; for every z > 0, (3) and xn < x hold for infinitely many

n , n

are also dense and they have the cardinality of the continuum, and X0 = x ∈ [0, 1]; there exists z > 0 such that (3) holds only for finitely many n is of the first category. (ii) For any xn ∈ [0, 1] and zn → 0, with the possible exception of a null set, the unit interval [0, 1] can be decomposed into two sets: X3 = x ∈ [0, 1]; for every z > 0, (3) holds only for finitely many n , X4 = x ∈ [0, 1]; for every z > 0, (3) and xn > x hold for infinitely many n

and also (3) and xn < x hold for infinitely many n .

These (i) and (ii) follow from the Denjoy–Young–Saks theorem on Dini derivates (cf. [2, p. 30]) and from the fact that the Dini derivates of f (x) in (4) are D + f (x) = sup z > 0; (3) and xn > x hold for infinitely many n , D − f (x) = − sup z > 0; (3) and xn < x hold for infinitely many n . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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4. MAIN RESULTS At the beginning we reformulate (see Theorem 2 below) the following general theorem on limit points of xn , n = 1, 2, . . . (see [11, Theorem 2.1]). Theorem 1. Let x0 ∈ [0, 1] be a discontinuity point of a d.f. g(x) ∈ G((xn )∞ n=1 ) with a jump of size h. Then there exists a subsequence xnk of xn such that (i) limk→∞ xnk = x0 , (ii) d((nk )∞ k=1 ) = h, where d((nk )∞ k=1 ) is the upper asymptotic density of nk , k = 1, 2, . . . . In the opposite direction, (i) and (ii) imply that there exists g(x) ∈ G((xn )∞ n=1 ) such that g(x) has at x0 a jump of size ≥ h. ∞ Theorem 2. Let (xn )∞ n=1 be a sequence in [0, 1) such that the set G((xn )n=1 ) of all d.f.s of (xn )∞ n=1 contains only continuous d.f.s. Then for every sequence zn > 0, zn → 0, and every x ∈ [0, 1] we have d((nk )∞ k=1 ) = 0 if |x − xnk | < znk , k = 1, 2, . . . . ∞ For every u.d. sequence xn we have G((xn )∞ n=1 ) = {g(x)}, g(x) = x, and thus d((nk )k=1 ) = 0. Consider another example:

Example 1. The sequence xn = log n mod 1,

n = 1, 2, . . . ,

has the set of d.f.s

emin(x,u) − 1 1 ex − 1 ; u ∈ [0, 1] , + u G(xn ) = gu (x) = eu e e−1

(6)

and {log Nk } → u implies FNk (x) → gu (x). This set G((xn )∞ n=1 ) was found by A. Wintner [25]. Thus, by Theorem 2, if |x − {log nk }| < znk ,

k = 1, 2, . . . ,

then k →0 nk

for every sequence zn > 0, zn → 0.

Note that recently Y. Ohkubo [16] proved the following fact: Let pn , n = 1, 2, . . . , be the increasing sequence of all primes. The sequence log pn mod 1, n = 1, 2, . . . , has the same d.f.s as log n mod 1. Thus nk , |x − {log pnk }| < znk , k = 1, 2, . . . , satisfies nkk → 0, again. The next theorem follows immediately from a criterion of continuity of d.f.s in [11, Theorem 2.3]. be a sequence in [0, 1) and zn → 0. For h = 1, 2, . . . denote Theorem 3. Let (xn )∞ 1 N n=1 2 2πihx n . If ωh = lim supN →∞ N n=1 e H 1 ωh = 0, lim H→∞ H h=1

then for every x ∈ [0, 1] the sequence of indices nk , |x − xnk | < znk , has d((nk )∞ k=1 ) = 0. G. Myerson [14] (see [24, Sect. 1.8.10]) defined a sequence (xn )∞ n=1 in [0, 1) to be uniformly maldistributed if for every subinterval I ⊂ [0, 1) with positive length |I| > 0 we have both lim inf n→∞

#{i ≤ n; xi ∈ I} =0 n

and

lim sup n→∞

#{i ≤ n; xi ∈ I} = 1. n


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In [23] a uniformly maldistributed sequence (xn )∞ n=1 is characterized by the fact that the set ∞ G((xn )n=1 ) contains all one-step d.f.s cα (x). In the way of Theorem 1 for such (xn )∞ n=1 one can prove the following: Theorem 4. Let (xn )∞ n=1 be a uniformly maldistributed sequence in [0, 1). Then there exists a decreasing sequence zn > 0, n = 1, 2, . . . , zn → 0, such that for every x ∈ [0, 1] the sequence of all indices nk , |x − xnk | < znk , k = 1, 2, . . . , has the upper asymptotic density d((nk )∞ k=1 ) = 1. Theorem 4 follows directly from Theorem 5 below. To see this we introduce the following notation: Let (xn )∞ n=1 be a sequence in [0, 1). For every x ∈ [0, 1] denote ∞ δ(x, (xn )∞ n=1 ) = sup d((nk )k=1 ); xnk → x . Similarly, for each subinterval I ⊂ [0, 1] denote δ(I, (xn )∞ n=1 ) = d({n ∈ N; xn ∈ I}). ∞ Obviously I1 ⊂ I2 implies δ(I1 , (xn )∞ n=1 ) ≤ δ(I2 , (xn )n=1 ), and if x is an interior point of I, then ∞ ∞ δ(x, (xn )n=1 ) ≤ δ(I, (xn )n=1 ). In addition, if (zn )∞ n=1 is any sequence of positive numbers converging to 0, denote ∞ γ x, (xn )∞ n=1 , (zn )n=1 = d {n ∈ N; |x − xn | < zn } . ∞ ∞ Evidently γ(x, (xn )∞ n=1 , (zn )n=1 ) ≤ δ(x, (xn )n=1 ) holds. The following theorem shows that this upper bound can be achieved at all points of the interval [0, 1]. Theorem 5. Let xn be a sequence in [0, 1). Then there exists a decreasing sequence zn → 0 ∞ ∞ such that γ(x, (xn )∞ n=1 , (zn )n=1 ) = δ(x, (xn )n=1 ) for every x ∈ [0, 1]. Proof. Let εn → 0 be a decreasing sequence and qn → 1 be an increasing sequence of positive numbers. Put n0 = 1 and define by induction an increasing sequence of integers nk as follows. Assume that the numbers n0 , . . . , nk−1 have already been defined. Cover [0, 1] by open intervals I1k , I2k , . . . , Iskk of length εk . Then for every i = 1, 2, . . . , sk there exists an integer mki > nk−1 such that #{n ≤ mki ; xn ∈ Iik } ≥ qk δ(Iik , xn ). (7) mki

Define nk = max{mki ; i = 1, 2, . . . , sk }. Now, having nk constructed, define zn to be any decreasing sequence such that znk+1 = εk for all k ∈ N. ∞ ∞ Let x ∈ [0, 1]. Since γ(x, (xn )∞ n=1 , (zn )n=1 ) ≤ δ(x, (xn )n=1 ), it is sufficient to prove that γ(x, xn , zn ) ≥ δ(x, xn ). For every k ∈ N there exists ik ≤ sk such that x ∈ Iikk . Denote k J = ∞ k=1 {n ∈ (nk−1 , nk ]; xn ∈ Iik }. By the choice of zn the inequalities |x − xj | < zj hold for all j ∈ J. Thus, using (7), we have k k ∞ ∞ ∞ δ(x, (xn )∞ n=1 ) ≤ lim inf δ(Iik , xn ) = lim inf qk δ Iik , (xn )n=1 ≤ d(J) ≤ γ x, (xn )n=1 , (zn )n=1 , k→∞

k→∞

which proves the theorem. Remark 1. J.A. Fridy [5] called the point x with δ(x, (xn )∞ n=1 ) > 0 a statistical limit point of . In [11] it is proved that the set a given sequence (xn )∞ n=1 x ∈ [0, 1]; δ(x, (xn )∞ n=1 ) > 0 is an Fσ -set and, vice versa, for any given Fσ -set there exists a related sequence (xn )∞ n=1 . On the ∞ other hand, the set G((xn )n=1 ) of d.f.s has the following fundamental properties (see [24, Sect. 1.7]) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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for every sequence (xn )∞ n=1 in [0, 1): (i) G((xn )∞ n=1 ) is non-empty, (ii) G((xn )∞ n=1 ) is closed, and (iii) G((xn )∞ n=1 ) is connected in the metric ⎛ d(g1 (x), g2 (x)) = ⎝

1

⎞1/2 (g1 (x) − g2 (x))2 dx⎠

.

0

Conversely, for any non-empty set H of d.f.s there exists a sequence xn ∈ [0, 1], n = 1, 2, . . . , such that G((xn )∞ n=1 ) = H if and only if H satisfies (i), (ii) and (iii). As we have mentioned in Theorem 1, {x ∈ [0, 1]; δ(x, xn ) > 0} = x ∈ [0, 1]; there exists g(x) ∈ G((xn )∞ n=1 ), x is a discontinuity point of g(x) . ∞ Thus, various examples of G((xn )∞ n=1 ) in [24] gave various examples of {x ∈ [0, 1]; δ(x, (xn )n=1 ) > 0}. G. Myerson [14] gave the following example of a maldistributed sequence: Example 2. The sequence

xn = {log log n},

n = 2, 3, . . . ,

has the set of d.f.s, by [7], G((xn )∞ n=1 ) = {cα (x); α ∈ [0, 1]} ∪ {hβ (x); β ∈ [0, 1]}, where FNk (x) → cα (x) if and only if {log log Nk } → α, and FNk (x) → hβ (x) if and only if {log log Nk } → 0 and ee ee

[log log Nk ]

[log log Nk ]+{log log Nk }

→ 1 − β.

In the following, for such a sequence xn , we find a concrete example of zn satisfying Theorem 4: Let (i) xn = {log log n}, k k+1 , k = 0, 1, 2, . . . , (ii) zn = Zk for n ∈ ee , ee 1 (iii) Zk = kc where c > 0 is an arbitrary large constant, and K+x+ZK for K = 1, 2, . . . . (iv) NK = ee Then, for every x ∈ [0, 1], the sequence of all indices nk , |x − xnk | < znk , k = 1, 2, . . . , has the upper asymptotic density d((nk )∞ k=1 ) = 1. More precisely, {n ≤ NK ; |x − xn | < zn } =1 K→∞ NK

(8)

lim

for every x ∈ [0, 1]. Let us prove this. We have |x − xn | < zn

xn ∈ (x − zn , x + zn ). (9) k k+1 , k = 0, 1, 2, . . . . Then by (9) Now, define zn as a constant zn = Zk for all integers n ∈ ee , ee

⇔

k k+1 k k+1 ; |x − xn | < zn = n ∈ ee , ee ; xn ∈ (x − Zk , x + Zk ) . n ∈ ee , ee PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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K K+1 Let NK be an integer from ee , ee , and assume that the sequence Zk , k = 0, 1, 2, . . . , is non-increasing. Then by (10) {n ≤ NK ; |x − xn | < zn } ⊃ {n ≤ NK ; xn ∈ (x − ZK , x + ZK )}. Thus #{n ≤ NK ; xn ∈ (x − ZK , x + ZK )} #{n ≤ NK ; |x − xn | < zn } ≥ NK NK ek ek+1 K−1 # n ∈ e ,e ; xn ∈ (x − ZK , x + ZK ) = NK k=0 K K+1 # n ≤ NK ; n ∈ ee , ee , xn ∈ (x − ZK , x + ZK ) + . NK

(11)

(12)

eK−1

As K → ∞, we have e eK → 0 and thus the sum in (12) also tends to zero. Since e eK eK+1 K+x−ZK eK+x+ZK n ∈ e ,e ,e and xn ∈ (x − ZK , x + ZK ) ⇔ n ∈ ee , K+x+ZK and express the last term in (12) as we can put NK = ee eK+x+ZK eK+x−ZK − e +1 e . K+x+ZK e e

(13)

(14)

Omitting the integer parts in (14) leads to the form 1−

1 K+x+ZK (1−e−2ZK ) ee

(15)

.

By the Lagrange theorem, 1 − e−2ZK > 2ZK e−2ZK , and the lower bound of (15) converges to 1−

1 K+x−ZK (2Z ) K ee

→1

(16)

if ZK = K1c with an arbitrary large constant c > 0. Another example of a maldistributed sequence is in [23, Par. 18]: Example 3. Let

√ log2 n ] [ log2 n , n = 1, 2, . . . , xn = 1 + (−1) where [x] denotes the integral part and {x} the fractional part of x. Then G((xn )∞ n=1 ) = {cα (x); α ∈ [0, 1]}. Theorem 5 can also be applied in the following [23, Par. 12, Theorem]: Example 4. For any sequence (xn )∞ n=1 in [0, 1) we have G((xn )∞ n=1 ) ⊂ {cα (x); α ∈ [0, 1]}

⇔

N 1 |xm − xn | = 0. N →∞ N 2 m,n=1

lim

∞ Moreover, if G((xn )∞ n=1 ) ⊂ {cα (x); α ∈ [0, 1]}, then G((xn )n=1 ) = {cα (x); α ∈ I}, where I is a closed subinterval of [0, 1] which can be found as N N 1 1 xn , lim sup xn , I = lim inf N →∞ N N N →∞ n=1 n=1


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and the length |I| of I can also be found as M N 1 |xm − xn |. |I| = lim sup M,N →∞ M N m=1 n=1

By Theorem 5 there exists a sequence (zn )∞ n=1 , zn > 0, zn → 0, such that for nk , |x − xnk | < znk , ∞ ) = 1 if x ∈ I and d((n ) / I. we have d((nk )∞ k k=1 ) = 0 if x ∈ k=1 In the following we study not only the upper asymptotic density d(nk ) of nk , |x − xnk | < znk , but also the possible limits #{n ≤ Nk ; |x − xn | < zn } , lim k→∞ Nk which better fit the terminology of d.f.s of xn . To do this we use a theory in [24]. Theorem 6. Let xn ∈ [0, 1) be a sequence with a d.f. g(x); i.e., there exist N1 < N2 < . . . such that #{n ≤ Nk ; xn ∈ [0, x)} → g(x) (17) FNk (x) = Nk for all x ∈ [0, 1]. Assume that g(x) has discontinuity jumps h(1) , h(2) , . . . at points x(1) , x(2) , . . . , respectively (infinitely many jumps are admissible). Then we can find a non-increasing sequence zn > 0, zn → 0, such that #{n ≤ MK ; |x(i) − xn | < zn } = h(i) K→∞ MK

(18)

lim

for every i = 1, 2, . . . . Here MK is a suitable subsequence of Nk . Proof. We shall construct sequences εK , ZK and MK , K = 0, 1, 2, . . . , such that (i) εK > 0, εK → 0 as K → ∞; (ii) ZK > 0, ZK is non-increasing, ZK → 0; (iii)

MK−1 MK

→ 0 and we put zn = ZK for n ∈ [MK−1 , MK ).

The sequence ZK has the following additional properties: (iv) h(i) + εK ≥ g(x(i) + ZK ) − g(x(i) − ZK ) for i = 1, 2, . . . , K; (v) ZK are sufficiently small so that the intervals (i) i = 1, 2, . . . , K, x − ZK , x(i) + ZK , are disjoint (see Fig. 1). By Fig. 1 we see (xn − ZK , 1 − xn − ZK ) ∈ [0, x(i) ) × [0, 1 − x(i) )

⇔

|x(i) − xn | < ZK .

(19)

Now we put MK = Nk for those k for which g(x(i) + ZK ) − g(x(i) − ZK ) + εK > FNk (x(i) + ZK ) − FNk (x(i) − ZK ) > g(x(i) + ZK ) − g(x(i) − ZK ) − εK

(20)

where ZK has been fixed. From the second inequality in (20) it follows that #{n ∈ [MK−1 , MK ); xn ∈ [x(i) − ZK , x(i) + ZK ]} MK−1 + > h(i) − εK , MK MK PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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DIOPHANTINE APPROXIMATION GENERALIZED √ 2ZK

203

(xn , 1 − xn ) (x(i) , 1 − x(i) )

1 − x(i) (xn − ZK , 1 − xn − ZK )

x(i) − ZK x(i) x(i) + ZK Fig. 1.

and from the first inequality in (20) and (iv) we have h(i) + 2εK >

#{n ∈ [MK−1 , MK ); xn ∈ [x(i) − ZK , x(i) + ZK ]} . MK

(22)

Applying (19) in (21) and (22), we get #{n ∈ [MK−1 , MK ); |x(i) − xn | < zn = ZK } #{n ∈ [MK−1 , MK ); xn ∈ [x(i) − ZK , x(i) + ZK ]} = MK MK (23) and consequently #{n ≤ MK ; |x(i) − xn | < zn } #{n ∈ [MK−1 , MK ); |x(i) − xn | < zn } ≤ MK MK #{n ∈ [MK−1 , MK ); |x(i) − xn | < zn } MK−1 ≤ + MK MK for every i = 1, 2, . . . and for K → ∞, which implies (18). ∞ ∞ To study D((xn )∞ n=1 , (zn )n=1 ) for the sequence xn , n = 1, 2, . . . , we can use not only G((xn )n=1 ) but also the discrepancy, dispersion or distances between the points of xn . More precisely, define dn and d∗n as follows: Reorder x1 , x2 , . . . , xn to the non-decreasing sequence xi1 ≤ xi2 ≤ . . . ≤ xin and denote dn =

d∗n =

max (xij+1 − xij ),

1≤j≤n−1

min (xij+1 − xij ).

1≤j≤n−1

They are used in the following Theorems 7 and 8. Theorem 7. Let xn , n = 1, 2, . . . , be an everywhere dense sequence in [0, 1], and let zn > 0, zn → 0, be a non-increasing sequence. (A) Suppose that cdn ≤ zn

for

(24)

n = 1, 2, . . .

holds with some positive constant c. Then |x − xn | < zn holds for infinitely many n for almost all ∞ x ∈ [0, 1]; i.e., |D((xn )∞ n=1 , (zn )n=1 )| = 1. (B) Suppose that ndn ≤ c < ∞ holds with some positive constant c. ∞ |D((xn )∞ n=1 , (zn )n=1 )| = 1.

for

n = 1, 2, . . .

Then the divergence of the series


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n=1 zn

2012

(25) implies that


204

Is ( 0

xij

Il )

(

)

xij+1

1

Fig. 2.

Proof. We use the method described in [19]. Denote Jn = (xn − zn , xn + zn ),

(26)

n = 1, 2, . . . .

Assume by contradiction that

D (xn )∞ , (zn )∞ < 1. n=1 n=1 ∞ ∞ ∞ Since D((xn )∞ n=1 , (zn )n=1 ) = k=1 n=k (Jn ∩ [0, 1]), there exists k such that ∞ (Jn ∩ [0, 1]) < 1. n=k

For simplicity, let k = 1. We express X = ∞ n=1 (Jn ∩ [0, 1]) as the sum of intervals Ii , i = 1, 2, . . . , such that (i) X = ∞ i=1 Ii , Ii ∩ Ij = ∅ for i = j, Ii are maximal intervals and (ii) |I1 | ≥ |I2 | ≥ . . . (zero intervals also admissible), and denote (iii) kn = the number of intervals Ii , i = 1, 2, . . . , intersecting {x1 , x2 , . . . , xn }. Now, assume that (iv) X = ∞ n=1 Jn ⊂ [0, 1], and the monotonicity z1 ≥ z2 ≥ . . . implies (v) |J1 | ≥ |J2 | ≥ . . . . From (i)–(v) it follows that |Jn | ≤ |Ikn |

for n = 1, 2, . . . .

(27)

Proof of (27). Let Ij be the interval from I1 , I2 , . . . for which j is the maximum of all i for which the interval Ii intersects x1 , x2 , . . . , xn . Let xi ∈ Ij , 1 ≤ i ≤ n. Since kn ≤ j, the monotonicity (ii) implies |Ij | ≤ |Ikn |. The monotonicity (v) implies |Jn | ≤ |Ji | and xi ∈ Ji ⊂ Ij ; then |Ji | ≤ |Ij |, which gives (27). εi Proof of (A). The convergence of ∞ i=1 |Ii | and the monotonicity of |Ii | imply |Ii | = i , εi → 0 as i → ∞, and (27) gives εk (28) |Jn | ≤ n kn for all n. of pairs (xij+1 , xij ) lying in different We see that kn defined in (iii) coincide with the numbers intervals I1 , I2 , . . . (see Fig. 2). Thus, using their sum (xij+1 − xij ), we have (xij+1 − xij ) 1 − |X| − εn ≥ (29) kn ≥ dn dn where εn = xi1 + (1 − xin ), and the density of xn implies εn → 0. Then by (28) we have 2zn = |Jn | ≤

dn εkn 1 − |X| − εn

(30)

for n = 1, 2, . . . . Thus by (30) zn ≤ ε∗n dn , where ε∗n → 0, which is a contradiction to the assumption cdn ≤ zn for all n. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proof of (B). Inputting the assumption ndn ≤ c < ∞ into (29), we have kn ≥ c n for all n, and inequality (27) gives |Jn | ≤ |Ic n |

∞

for n = 1, 2, . . . .

(31)

∞

The ∞ convergence i=1 |Ii | < ∞ and (31) directly imply the convergence n=1 |Jn | < ∞ and then n=1 zn < ∞, a contradiction. Finally, opposite to (iv), we examine the case (vi) X = ∞ n=1 Jn ⊂ [0, 1]. We again express X = ∞ i=1 Ii , where Ii are pairwise disjoint, |I1 | ≥ |I2 | ≥ . . . , and assume that 0 ∈ Ii , 1 ∈ Ij , and both intervals intersect x1 , x2 , . . . , xn . Here we again use the density of xn , n = 1, 2, . . . , in [0, 1]. Let I be the minimal interval containing X. Then (29) has the form kn ≥

|I| − |X| dn

for n = 1, 2, . . . .

(32)

Remark 2. From the definition of discrepancy Dn in Section 2 we see that dn ≤ Dn ; thus (24) can be replaced by cDn ≤ zn

for n = 1, 2, . . . .

1 for all n and there exists an absolute constant c > 0 such that Dn > c logn n Furthermore, Dn ≥ 2n for infinitely many n (W.M. Schmidt, 1972, cf. [24, pp. 1–43]). Remark 3. For a measure of denseness of a sequence (xn )∞ n=1 in [0, 1) H. Niederreiter [15] introduced dn = sup min |x − xi |, x∈[0,1] 1≤i≤n

which is called the dispersion of (xn )∞ n=1 . Here dn and dn have a similar meaning. Reordering x1 , x2 , . . . , xn to the non-decreasing sequence xi1 ≤ xi2 ≤ . . . ≤ xin , we have (see [24, Sect. 1.10.11])

1 max (xi − xij ), xi1 , 1 − xin . dn = max 2 1≤j≤n−1 j+1 Thus dn ≤ 2dn . H. Niederreiter [15] proved that the limit lim ndn =

n→∞

1 2 log 2

is the lowest possible. The following example was given by I. Ruzsa, see [15]. Example 5. Consider the sequence x1 = 1,

xn =

log(2n − 3) mod 1 for n = 2, 3, . . . . log 2

Then

log n − log(n − 1) for n = 2, 3, . . . . dn = 2 log 2 By the previous Theorem 7(B), if ∞ n=1 zn = ∞, then for almost all x ∈ [0, 1], |x − xn | < zn holds for infinitely many n = nk , but by Theorem 2 in all cases the asymptotic density d((nk )∞ k=1 ) = 0. Another application of Theorem 7(B) is the sequence of dyadic numbers. Example 6. Let (xn )∞ n=1 be the sequence of all dyadic numbers ordered as 1 1 3 1 3 5 7 1 0, , , , , , , , , . . . . 2 4 4 8 8 8 8 16 PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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206

I1

I3

0

( )( ) (

I2

)( )( ) ( I2

I1

. . . Il 1 )( ) ( )( ) I4 . . . Il+1

)( )( ) ( I3 Fig. 3.

For 2k−1 < n ≤ 2k we have dn =

max (xij+1 − xij ) ≤

1≤j≤n−1

1 . 2k−1

1 ≤ 2 for n = 1, 2, . . . . Then ndn ≤ 2k 2k−1

Theorem 8. Let xn , n = 1, 2, . . . , be a u.d. sequence in [0, 1], and let zn > 0, zn → 0, be a non-increasing sequence. Assume that nd∗n ≥ c > 0 holds some positive constant c. with D (xn )∞ , (zn )∞ = 1. n=1 n=1

for

(33)

n = 1, 2, . . .

Then the divergence of the series

∞

n=1 zn

implies that

Proof. Again, let the sequence xn ∈ [0, 1), n = 1, 2, . . . , be covered by pairwise disjoint open intervals Ii , i = 1, 2, . . . , ordered as |I1 | ≥ |I2 | ≥ . . . , kn = the number of Ii containing some term of x1 , x2 , . . . , xn , and d∗n =

min (xij+1 − xij ),

1≤j≤n−1

where xi1 ≤ xi2 ≤ . . . ≤ xin .

be component intervals of [0, 1] \ li=1 Ii (see Fig. 3) and the sum (xij+1 − xij ) Let I1 , I2 , . . . , Il+1 l+1 contain all pairs (xij , xij+1 ) lying in the same interval Ii and contained in j=1 Ij . Then } − S, (34) kn ≥ #{i ≤ n; xi ∈ I1 ∪ I2 ∪ . . . ∪ Il+1 − xij ). Using the uniform distribution of xn , diswhere S is the number of summands of (xij+1 ∗ crepancy Dn of x1 , x2 , . . . , xn , (xij+1 − xij ) ≤ ∞ i=l+1 |Ii | and applying the minimum dn , we find

kn ≥

1−

l

∞

|Ii | n − (l + 1)nDn −

i=1

i=l+1 |Ii | d∗n

(35)

n. Then, we can apply for an arbitrary l = 1, 2, . . . . Assuming nd∗n ≥ c > 0, we have kn ≥ c ∞ ∞ the chain from the proof ∞following ∞ 7: If D (xn )n=1 , (zn )n=1 < 1, then there exists ∞ of Theorem n=k Jn < 1; express i=1 Ii , Ii pairwise disjoint, |Ii | monotone; this implies it as n=k Jn = ∞ |J | < ∞; then |Jn | ≤ |Ic n | and thus ∞ n=1 n n=1 zn < ∞, a contradiction.

Theorem 8 can be applied to the sequence xn = θn mod 1, n = 1, 2, . . . , for irrational θ with bounded partial quotients. Example 7. Let θ = [a0 ; a1 , a2 , . . .] be the continued fraction expansion of an irrational θ. Denote pk = [a0 ; a1 , a2 , . . . , ak ], qk

rk = [ak ; ak+1 , ak+2 , . . .],

qk = qk−1 ak + qk−2 ,

0 < {θi1 } < {θi2 } < . . . < {θin } < 1, d∗n =

min ({θij+1 } − {θij }).

1≤j≤n−1


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If qk−1 ≤ n < qk , then d∗n ≥ |θqk − pk | =

1 qk rk+1 +

Assume that ai ≤ K for i = 1, 2, . . . . Then we have qk−1 1 nd∗n ≥ qk−1 = qk−2 qk rk+1 + qk ak + qk−1 rk+1 +

qk−1 . qk

qk−1 qk

≥

1 . (K + 1)(K + 2)

ACKNOWLEDGMENTS This work was supported by VEGA 1/0753/10 and 2/0206/10. REFERENCES 1. D. Berend and A. Dubickas, “Good Points for Diophantine Approximation,” Proc. Indian Acad. Sci., Math. Sci. 119 (4), 423–429 (2009). 2. A. M. Bruckner and J. L. Leonard, “Derivatives,” Am. Math. Mon. 73 (4, Part 2), 24–56 (1966). 3. R. J. Duffin and A. C. Schaeffer, “Khintchine’s Problem in Metric Diophantine Approximation,” Duke Math. J. 8, 243–255 (1941). 4. P. Erd˝ os, “On the Distribution of the Convergents of Almost All Real Numbers,” J. Number Theory 2, 425–441 (1970). 5. J. A. Fridy, “Statistical Limit Points,” Proc. Am. Math. Soc. 118 (4), 1187–1192 (1993). 6. P. X. Gallagher, “Metric Simultaneous Diophantine Approximation. II,” Mathematika 12, 123–127 (1965). 7. R. Giuliano Antonini and O. Strauch, “On Weighted Distribution Functions of Sequences,” Unif. Distrib. Theory 3 (1), 1–18 (2008). 8. G. Harman, Metric Number Theory (Clarendon Press, Oxford, 1998), London Math. Soc. Monogr. 18. 9. A. Haynes, A. Pollington, and S. Velani, “The Duffin–Schaeffer Conjecture with Extra Divergence,” arXiv: 0811.1234v3 [math.NT]. 10. A. Khintchine, “Einige S¨ atze u ¨ber Kettenbr¨ uche, mit Anwendungen auf die Theorie der Diophantischen Approximationen,” Math. Ann. 92, 115–125 (1924). ˇ at, and O. Strauch, “On Statistical Limit Points,” Proc. Am. Math. Soc. 129 (9), 11. P. Kostyrko, M. Maˇcaj, T. Sal´ 2647–2654 (2001). 12. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (J. Wiley & Sons, New York, 1974; Dover Publ., Mineola, NY, 2006). 13. J. Lesca, “Sur les approximations diophantiennes ` a une dimension,” Doctoral Thesis (Univ. Grenoble, 1968). 14. G. Myerson, “A Sampler of Recent Developments in the Distribution of Sequences,” in Number Theory with an Emphasis on the Markoff Spectrum, Provo, UT, 1991 (M. Dekker, New York, 1993), Lect. Notes Pure Appl. Math. 147, pp. 163–190. 15. H. Niederreiter, “On a Measure of Denseness for Sequences,” in Topics in Classical Number Theory: Colloq. Budapest, 1981 , Ed. by G. Hal´ asz (North-Holland, Amsterdam, 1984), Vol. 2, Colloq. Math. Soc. J. Bolyai 34, pp. 1163–1208. 16. Y. Ohkubo, “On Sequences Involving Primes,” Unif. Distrib. Theory 6 (2), 221–238 (2011). 17. M. Reversat, “Un résultat de forte eutaxie,” C. R. Acad. Sci. Paris A–B 280, A53–A55 (1975). 18. V. G. Sprindzuk, Metric Theory of Diophantine Approximations (Nauka, Moscow, 1977; V.H. Winston & Sons, Washington, DC; J. Wiley & Sons, New York, 1979). 19. O. Strauch, “Duffin–Schaeffer Conjecture and Some New Types of Real Sequences,” Acta Math. Univ. Comenianae 40–41, 233–265 (1982). 20. O. Strauch, “A Coherence between the Diophantine Approximations and the Dini Derivates of Some Real Functions,” Acta Math. Univ. Comenianae 42–43, 97–109 (1983). 21. O. Strauch, “Two Properties of the Sequence nα (mod 1),” Acta Math. Univ. Comenianae 44–45, 67–73 (1984). 22. O. Strauch, “L2 Discrepancy,” Math. Slovaca 44, 601–632 (1994). 23. O. Strauch, “Uniformly Maldistributed Sequences in a Strict Sense,” Monatsh. Math. 120, 153–164 (1995). ˇ Porubsk´ 24. O. Strauch and S. y, Distribution of Sequences: A Sampler (Peter Lang, Frankfurt am Main, 2005). 25. A. Wintner, “On the Cyclical Distribution of the Logarithms of the Prime Numbers,” Q. J. Math., Oxford Ser. 6, 65–68 (1935).

This article was submitted by the authors in English PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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