CONVERGENCE RATES OF SUBSERIES PAOLO LEONETTI
Abstract. Let (xn ) be a positive real sequence decreasing to 0 such that the P series x n n is divergent and lim inf n xn+1 /xn > 1/2. We show that there exists a constantPθ ∈ (0, 1) such that, for each ` > 0, there is a subsequence (xnk ) for which k xnk = ` and xnk = O(θk ).
1. Introduction. P Given a real sequence x = (xn ) with divergent series n xn , let A(x) be its achievement set, that is, the set of sums of convergent subseries P A(x) = ` ∈ R : k≥1 xnk = ` for some subsequence (xnk ) . In this regard, it is known that “almost all” subseries (both in the measuretheoretic sense and the categorical sense) diverge, see [4, 16, 18]. Related results in the context of filter convergence have been recently proved in [2, 11, 12]. There is a large literature on achievement sets; see, e.g., [7, 8, 10, 14, 15]. Achievement sets are also known as subsum sets: it is known, for instance, that if (xn ) is a sequence whose terms form a conditionally convergent series then A(x) = R; see [10, Corollary 1.4]. In addition, if (xn ) is a sequence converging to 0 then A(x) is one of the following three possibilities: a finite union of (nontrivial) compact intervals, a Cantor set, or a “symmetric Cantorval” (that is, a Cantorlike set with both trivial and nontrivial components); see [14, p. 870]. Motivations for the investigation of these sets come from, among other areas, measure theory and fractal geometry; see [5] and [13], respectively. Of special interest have been P specific subseries of the harmonic series n n1 ; see, e.g., [1, 3, 17, 20, 21]. Motivated by the study of achievement sets in the context of Banach spaces [6], Jacek Marchwicki posed the following question during the open problem session of the 46th Winter School in Abstract Analysis (Czech Republic, 2018): Question 1. Fix a real ` > 0. Does there exist a set of distinct primes P such that X1 X 1 = ` and √ < ∞? p p p∈P p∈P 2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 26A12. Key words and phrases. Series, subseries, achievement set, rate of convergence.
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Paolo Leonetti
Informally, the question above aims to establish whether there is a relationship between numbers ` in the achievement set A(x), where each xn is the reciprocal of P the nth prime, and the growth rate of the subsequences (xnk ) for which k xnk = `. In this respect, we are going to prove that there exists a sequence (xnk ) of this type that decays at least exponentially. As a consequence of our main result (Theorem 3 below), we give an affirmative answer to Question 1, and more, by proving the following corollary to it. Corollary 2. For each ` > 0 and for each θ > 0, there exists an increasing sequence of primes (pk ) and a constant c > 0 such that X 1 = ` and pk ≥ c θk for all k. p k k P 1 In particular, k pα < ∞ for all α > 0. k P A related result on the rate of growth of the partial sums ( ni=1 xi ) can be found in [9]. Hereafter, given a real sequence (an ) and a function f : N → (0, ∞), the notation an = O(f (n)) stands for lim supn→∞ an /f (n) < ∞. The main result of this note is the following. P Theorem 3. Let (xn ) be a positive nonincreasing real sequence such that n xn = +∞, limn→∞ xn = 0, and L := lim inf n→∞
xn+1 1 > . xn 2
(1)
Then there exists a constantPθ ∈ (0, 1) such that, for each ` > 0, there is a subsequence (xnk ) for which k xnk = ` and xnk = O(θk ). It is worth noting that it is well known that every sum can be achieved, i.e., A(x) = (0, ∞); see [19, Theorem 1.1] and [14, p. 863]. Hence the novelty here concerns the rate of convergence of subseries, which is asymptotically (at most) geometric. In addition, these rates are bounded uniformly for all desired subsums. As it follows from the proof of Theorem 3, an estimate on the value of θ can be found if L is not small: Corollary 4. With the same hypotheses as in Theorem 3, suppose that √ xn+1 5−1 L := lim inf > (≈ 0.618034). n→∞ xn 2 Then, for each ` > 0 and for each ε > 0, there exists a subsequence (xnk ) for √ P k which k xnk = ` and xnk = O 1+ε−L . In particular, if L = 1, we obtain the following result (we omit details).
Convergence Rates of Subseries
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P Corollary 5. Let (xn ) be a positive nonincreasing real sequence such that n xn = +∞, limn→∞ xn = 0, and limn→∞ xxn+1 = 1. Then, for each ` > 0 and for each n P θ > 0, there exists a subsequence (xnk ) for which k xnk = ` and xnk = O(θk ). Finally, we have the following easy consequence. Corollary 6. With the hypotheses of Theorem P P 3,α for each ` > 0 there exists a subsequence (xnk ) such that k xnk = ` and k xnk < ∞ for all α > 0. Proof. If α < 1, the claim follows easily by Theorem 3 (we omit details). Hence, let us suppose that α ≥ 1. Since x is convergent to 0, there exists an integer N ≥ 1 such that xn < 1 for all n ≥ N . Therefore X X xαnk ≤ N kxkα∞ + xnk ≤ O(1) + ` < ∞. k≥1
k≥N
Proof of Corollary 2. Thanks to the prime number theorem, the nth prime is equal to n log n(1 + o(1)), so that the inequality (1) holds with L = 1. The result follows by Corollary 5 and Corollary 6. 2. Further Proofs. Proof of Theorem 3. First, suppose that α ∈ (0, 1) and fix ` > 0. Let a1 be the smallest integer a such that xa < `, which is well-defined P because limn→∞ xn = 0. b Also, let b1 be the greatest integer b ≥ a1 such that n=a1 xn < `. Define Pb1 S1 := j=a1 xj . Then recursively construct the increasing sequences of integers Pn xj , in (an ) and (bn ), as well as the related sequence of real numbers Sn := bj=a n the analogous way: (i) Let an+1 be the smallest integer a > bn such that xa < ` −
n X
Si ;
i=1
(ii) Let bn+1 be the greatest integer b ≥ an+1 such that b X
xj < ` −
j=an+1
n X
Si .
i=1
Note that, by the maximality of bn , we S have an+1 ≥ bn + 2. Denoting byP(nk ) the increasing enumeration of (infinite) set n [an , bn ], it easily follows that k xnk = `. Indeed, n X 0≤`− Si ≤ xbn +1 → 0 i=1
as n → ∞.
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Paolo Leonetti
˜ := L − ε > 1 , and set At this point, fix ε > 0 sufficiently small so that L 2 N = N (ε) > 1 such that ˜ xn xn+1 ≥ L Pn Pn−1 for all n ≥ N . Note that i=1 Si < ` ≤ i=1 Si + xan −1 for all n > 1, implying that ˜ + ··· + L ˜ κn xan −1 xan −1 > Sn = xan + · · · + xbn ≥ L (2) ˜ > 1 , it easily follows for all n ≥ N , where κn := bn − an + 1. Considering that L 2 that K := lim sup κn < ∞. (3) n→∞ P Pn−1 With similar reasoning, we have n+1 i=1 Si < ` ≤ i=1 Si + xan −1 for all n > 1. This implies that κn+1 xbn+1 ≤ Sn+1 < xan −1 − Sn ≤ xan −1 − xan ˜ ≤ xbn−1 (1 − L). ˜ ≤ xan −1 (1 − L) ˜ n/2 . Considering that there exists an integer M ≤ 0 Therefore, xbn = O (1 − L) such that nk ≥ bbk/Kc + M for all large k, we conclude that ˜ k/K+M 2 xnk ≤ xbbk/Kc +M ≤ xbbk/Kc+M = O (1 − L) = O(θk ), (4) ˜ 1/2K . This completes the proof. where θ := (1 − L)
Proof of Corollary 4. With the same notation as in the proof above, fix a suffi√ 5−1 ˆ ciently small ε > 0 such that L := L − ε > 2 . Then inequality (2) holds only if κn = 1 for all n ≥ 2; hence K = 1. The claim follows by the estimate (4). 2.1. Acknowledgment. The author is grateful to the editor and the two anonymous referees for their remarks that allowed a substantial improvement of the presentation. References 1. R. Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly 86 (1979), no. 5, 372–374. 2. M. Balcerzak, M. Poplawski, and A. Wachowicz, Ideal convergent subseries in Banach spaces, Quaest. Math., to appear. 3. R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly 78 (1971), 864–870. 4. R. C. Buck, Limit points of subsequences, Bull. Amer. Math. Soc. 50 (1944), 395–397. 5. C. Ferens, On the range of purely atomic probability measures, Studia Math. 77 (1984), no. 3, 261–263. 6. S. Glab and J. Marchwicki, Levy-Steinitz theorem and achievement sets of conditionally convergent series on the real plane, J. Math. Anal. Appl. 459 (2018), no. 1, 476–489. 7. J. A. Guthrie and J. E. Nymann, The topological structure of the set of subsums of an infinite series, Colloq. Math. 55 (1988), no. 2, 323–327.
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8. J. D. Hill, Some theorems on subseries, Bull. Amer. Math. Soc. 48 (1942), 103–108. 9. E. H. Johnston, Rearrangements of divergent series, Rocky Mountain J. Math. 13 (1983), no. 1, 143–153. 10. R. Jones, Achievement sets of sequences, Amer. Math. Monthly 118 (2011), no. 6, 508–521. 11. P. Leonetti, Thinnable ideals and invariance of cluster points, Rocky Mountain J. Math., to appear. 12. P. Leonetti, H. Miller, and L. Miller-Van Wieren, Duality between measure and category of almost all subsequences of a given sequence, Period. Math. Hungar., to appear. 13. M. Mor´ an, Dimension functions for fractal sets associated to series, Proc. Amer. Math. Soc. 120 (1994), no. 3, 749–754. 14. Z. Nitecki, Cantorvals and subsum sets of null sequences, Amer. Math. Monthly 122 (2015), no. 9, 862–870. 15. J. E. Nymann and R. A. S´ aenz, On a paper of Guthrie and Nymann on subsums of infinite series, Colloq. Math. 83 (2000), no. 1, 1–4. 16. M. B. Rao, K. P. S. B. Rao, and B. V. Rao, Remarks on subsequences, subseries and rearrangements, Proc. Amer. Math. Soc. 67 (1977), no. 2, 293–296. 17. T. Schmelzer and R. Baillie, Summing a curious, slowly convergent series, Amer. Math. Monthly 115 (2008), no. 6, 525–540. ˇ at, On subseries, Math. Z. 85 (1964), 209–225. 18. T. Sal´ ˇ , On subseries of divergent series, Mat. Casopis Sloven. Akad. Vied 18 (1968), 312– 19. 338. 20. A. D. Wadhwa, An interesting subseries of the harmonic series, Amer. Math. Monthly 82 (1975), no. 9, 931–933. , Some convergent subseries of the harmonic series, Amer. Math. Monthly 85 (1978), 21. no. 8, 661–663. ` “Luigi Bocconi”, Department of Statistics, Milan, Italy Universita E-mail address:
[email protected] URL: https://sites.google.com/site/leonettipaolo/
