Mišík, Ladislav

Žáčik, Tibor

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaTibor Žáčik; Ladislav MišíkA simplified formula for calculation of metric dimension of converging sequencesMathematica Slovaca, Vol. 55 (2005), No. 3, 363--372Persistent URL: http://dml.cz/dmlcz/130558Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2005Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathernatica Slovaca ©2005 _. __ ,~~nr-\ . . .«, _.,.,>, __,,- Mathematical Insti tute Math. SlOVaCa, 55 (2005), NO. 3, 363-372 Slovák Academy of Sciences A SIMPLIFIED FORMULA FOR CALCULATION OF METRIC DIMENSION OF CONVERGING SEQUENCES TlBOR Z A C I K * — L A D I S L A V M I S I K * * (Communicated by Lubica Hold) ABSTRACT. In [MISIK, L.—ZACIK, T.: A formula for calculation of metric dimension of converging sequences, Comment . Math . Univ. Carolin. 4 0 (1999), 393 401] a formula for calculation of metric dimension of convex sequences con-verging to 0 was derived. In this paper, the formula is studied in more details. A much simpler formula is derived and mutual relations between two quantities determining metric dimension are described. Introduction A frequent problem in mathematics is evaluation of the size of sets. An im-portant indicator is the dimension of a set. There are more different concepts of dimension. All kinds of dimensions are equal on "nice" sets, for example on open sets in W1. In this contribution we will consider the metric dimension. It is the only kind of dimension which can be positive on countable sets. In the paper, we deal with a calculation of the (upper) metric dimension of sequences of real numbers. According to the knowledge of the authors, the only method for calculation of the upper metric dimension was published by H a w k e s ([H], this method was based on result presented in [BT]), and there was no method for calculation of the lower metric dimension. In [MZ2], formulae for calculation both, the upper and lower metric dimensions, were derived. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : 54F50, 40A05. K e y w o r d s : metric dimension, box-counting dimension, limit capacity, Kolmogorov dimen-sion, Minkowski dimension, Bouligand dimension, entropy dimension, Hausdorff dimension, converging sequences, convex sequence, differentiable function. Partially supported by the grand VEGA 2/1140/21 and by GA CR 201/04/0381/2. 363 TIBOR lkt\K — LADISLAV Ml5IK The main result of the paper is to derive a simpler (in comparison with [MZ2]) formula of calculation the metric dimension of a class of real sequences defined by convex differentiable functions. The use of the formula covers all calculations made in [KA]. Preliminaries Let X be a totally bounded metric space. Lower metric dimension dim of totally bounded metric space X was introduced by P o n t r y a g i n and S n i r e l m a n in [PS] by t\og N(r,X) dim A = limmi -r->o+ — logr and upper metric dimension dim by K o l m o g o r o v and T i h o m i r o v in [KT] by dim X = hm sup . r^o+ - l o g r Here jN(r, X) denotes the least number of open balls of diameter r covering the space X. In the case dim A = dim A , we define the metric dimension of A" dim A = dim A . Metric dimension is sometimes called limit capacity, entropy dimension, box-counting dimension, Kolmogorov dimension, Minkowski dimension, and Bouli-gand dimension. We remark that t d A < h d A < d i m A <dim~A, where td denotes the topological dimension and hd the Hausdorff dimension. It is well known that there exist countable sets of positive metric dimension. There are even examples of sets whose metric dimension equals to that of the original space. One can think that countable sets of positive metric dimension are very rare and of complicated structure. In fact, as shown in [MZ1], the opposite is true. THEOREM. Every infinite totally bounded space X contains a converging se-quence {an}n=zl such that d i m K j ^ L i = d i m X We remark that an analogue of this theorem does not hold for lower metric dimension. 364 CALCULATION OF METRIC DIMENSION OF CONVERGING SEQUENCES In [MZ2] the concepts of convex sequence and its associate function are in­troduced. By a convex sequence (c-sequence, in short) we mean a decreasing sequence of positive real numbers converging to zero which is convex as a function denned on the set of all positive integers. Any C1 convex decreasing function / : M+ —> JR+ such that an = f(n) is called an associated function to the given convex sequence KKTV In the above mentioned paper the following formulae for calculation both upper and lower metric dimension of any convex sequence A = {an}n<>-1 by means of its associated function / are derived dim A = lim inf ^— i- , *^oo _ l o g ( - / ' ( ; r ) ) and ^(*-Ш) dim A = lim sup , , . (1) x^oo - l o g ( - / ' ( z ) ) Two corollaries of this formulae enable us to calculate the metric dimension under the existence of all limits occurring: _ l o g f * - ^ ) dirndl = dirndl = dirndl = lim ^ — - j — f (2) z^oo - log(-/ '(a;)J T 4 T I T A ( T lOgX . «. lOg(f(x)) ) dim A — dim A = dim A = max < lim —-—-—,,, x, , 1 — lim -—-—,,, „ > I z-+oo -log(-/'(x)) ' x->oo l Q g(-/ (^)) J (see [MZ2; Theorem, Corollary 1, Corollary 2]). In this paper these formulae are analysed in more details, and a much simpler formula is derived. Main results We will start with a simple generalization of [MZ2; Corollary 2]. PROPOSITION 1. Let A = {an}n<L1 be a c-sequence with an associate func­tion / . Then max < lim inf —-.—-—,., ,, , lim inf -—~ f (x) K < dirndl [ x^oo log(-/'(x)) ^^oo log _//(a;) J / r ^gx .. l o g ^ ) l - p - , < max < lim sup —;—-—,,, „ , lim sup ------ > = dim A . I ^^oo log(-rw) ' ^ o o l o s - 7 ^ ) J A proof of this proposition is omitted since it is similar to that of [MZ2; Corollary 2]. 365 TIBOR 2ACIK — LADISLAV MlSlK LEMMA 1. Let f be a decreasing convex function with lim f(x) = 0 and let x-±oo the inequalities 0 < a x0 . for some x0 > 0. Then lOK f(~) \0p / W b+1--^- < l iminf zI^-- < l i m s u p ^--- < m i n { b , l } . a *"•«> l o g 3 j % y *->°° log 377%) l o g _/!___ P r o o f . Since / (x ) < 1 for sufficiently large x, lim sup -—~/1(x) is bounded .r-»oo o g - / ' ( * ) from above by 1. The assumption a > 1 in (3), after integration similar to that used later to derive (5), leads to a contradiction. This allows us to consider only the case 0 < a xQ). (4) Integrating (4) in the interval (x, 00) gives 00 00 00 f -t-13-1 dt< f f(t) dt 0, /3 > 0. Hence, t—>-oo ^ < / ( r r ) < ^ (^>x0). (5) Rewriting (4) and (5) gives the forms X0+1 < -J(x) < X°+1' (4'} P*0 < TTT < ax" • (5') ZOO 366 CALCULATION OF METRIC DIMENSION OF CONVERGING SEQUENCES NT *• _- l o l ^ § k 1 l o g 7^T Now, we estimate -j—- — 1 --—ГЩ І 0 S - 7 7 - - ; f(x) ß 1 _ log a s " -Qgr/T^y _ log/jx log _r^+1 log 3j}-- - log _ra+1 ' _ a log x + log a lQg -//(„) _ /Jlogx + log/3 (/? + l)log_r lQg_7T(-) (a + l)log_r Letting x tend to infinity we obtain a l o g - ^ r l o g ^ f r 3 a - -• . - 6 -/ '(*) . ,. & -f'(x) / - , P 1 - - — - < l iminf ----- < l i m s u p ^- < 1 -0+1 _->oo log - - I — X_+TO l o g r y ^ y a +1 ' or, equivalently, returning to original values a and 6, lop- /(^) lop- f(x) 6 + l - - - < l i m i n f = ^ < l i m s u p _ £ _ _ . < 0 + 1 - • £ . (6) In the case 6 = 1 the inequalities (4') and (5'), needed for the lower estima-. l o g - — -tion of liminf -— f'^x) , can be rewritten to the form: x—>oo 1 O S -f'(x) - < _7щ < І" 1 . <"'.) 1 /(*) (5І) Now, /(*) l i m i n t 'o s ^ > l i m i „ f í i -l ° ' ^ > x-+oo Jog — * N _c->oo V l o g X o - / ' ( „ ) v = i _ i i m loga + crlogx x-+oo l ogX = l _ a = l _ _ _ _ i = 2 - I = 6 + l - * . a a a The lower estimation is now proved. The upper estimation is evidently true for b = 1. Now we prove the upper estimation in the case b < 1 using (5) and convexity of / . Fix x > _c0, and let x1 > x be such that a line / passing through points (x,/(_c)) and (x1 , ^_ - ) is the tangent line to the graph of H = ^—- at point ( x i P \ \XH (3 J ' SO X / ( x ) = - L _ - x ^ - Ҷ x - x . ) < x " Ҷ l + ì ) /З "^1 v~ ~1 У ^ - " 1 v~ ' ß> 367 I .Є., TIBOR ŽÁČIK — LADISLAV MIŠÍK > m i + y By convexity of / and (5), the gradient of line / is greater than that of the tangent to the graph of / at point (x,f(x)), so f'(x) < —x^~ , i.e., 1 /3+1 -n*)<Xl ' Therefore, by decreasing nominator and increasing denominator in -— n*) we obtain estimation i - r - i r L < i -l o S ^ F ( i ) From this, as x < x1, l o § 7 M , l o g ^ I , ^ l o g a : i - l o g ( l + i ) logx^ /H-l = 1 l imsup: -V--- < 1 - -x " ~ l o g - r - i - y - ^ lJ + 1 /J + l (/î + ^ l o g x j = 6. (7) As for a b , b provides a better upper log _/(*) bound of limsup -—~ / 1( x ) as the upper bound in (6). Z-+OO l 0 g -f'(x) This finishes the proof of lemma. D 368 CALCULATION OF METRIC DIMENSION OF CONVERGING SEQUENCES R e m a r k 1. Note that the upper estimation and its proof depend only on b. So the upper estimation can be applied also in the case when 0 < -— 1 x0 instead of (3). -/'(*) The following theorem provides a much simpler means of calculation of upper metric dimension as (1). THEOREM 1. Let A = {an}n=:1 be a c-sequence and let f be its associated function. Then : r— A v l o 6 £ dim A = hm sup -— . * ^ o o log 3/77^) P r o o f . Proof is a straightforward consequence of Proposition 1 and Lemma 1, where we take b = lim sup-— x + e for an arbitrary e > 0. z-+oo o g - / ' ( * ) • Similarly, the following theorem enables a simpler calculation of metric di-mension in comparison to (2). THEOREM 2. Let A = {an}^L1 be a c-sequence and let f be its associated function. Then dim A = lim =— *->«> log ^77^) provided the limit exists. P r o o f . Let lim -—°gx = r and r > 0. Then for each e, 0 < e < r, Z-+00 l°g • ^rjx) there is x0 such that logx •e < —,— <r + e for each x > x0. Lemma 1 then yields W /(*) w fW r + £ + l - ^ ^ < l i m i n f ^ 4 ^ - < l i m s u p ^ 4 ^ < r + £ . r-e ^ 0 0 l o g r 7 7 R x^oc l o g 3 7 7 R If e tends to zero, both sides of previous inequalities tend to r . Proposition 1 then completes the proof in the case r > 0. Let r = 0. Then for each e > 0 there is x0 such that logz 0 < - 5 _ _ < £ for each x > x0. By Lemma 1 and Remark 1, log^My log^fe - c o < lim inf - -1LJ- < lim sup --7-—-- < e or- V r^r\ I r\re ± ts^rr -*-.. x—>oo l o g r j r ^ x^oo log^T^y 369 TIBOR ZAClK — LADISLAV MlSlK Therefore, l0g f(*) l0g /(*) - o o < l iminf = ^ - - < l i m s u p = ^ - - < 0. ^ ° ° l o g ^ ( - ) *^°° ^ S ^ W From Proposition 1 j - — ^ f r ^S* r l°g -N*) dim A = max < lim sup —-}—(_f/( ,, , Aim sup __>-_ " log(-/'(^)) ' X_>J* log ^ hence dim_4 = 0. The theorem is proved. D = max <{ 0 , lim sup : æ^oo Examples 1) Let {an}^_i = { l /n Q } ^ = i i OL> I, with the associate function f(x) = x a Then ,. r i / a l r !og£ ,. logx d i m { l / n a } = lim —-—-(—-7—r = lim — . £->-oo — log( — J (X)) x^oo — log CTX a L ,. log X 1 = lim - ~ — — . z->oo — logCT + (a + 1) logX CT + 1 2) Let {an}^°__1 = { rn } ~ = l J 0 < r < 1, with the associate function / (x ) Then ,. f m r l Q g ^ T _ _ l o g z dim{r } = lim —-—-———- = lim —-— 3:̂ 00 — log(—f'(x)) x-^00 — log(—rx logr) l i m ____ = Q .r->oo —xlogr — log(— logr) 3) Let {an}^=1 = { 1 / l o g n } ^ with the associate function f(x) = (logx) Then r / , 1 T l O g X n. l O g X d i m { l / l o g n } = hm — — = lim — ^ 0 0 - l og ( - / ' ( x ) ) *->°o - log 7 ^ - " J logx logx = lim -5— = lim - —-— = 1 . x^oo log a;(log x) *->^ log x + 2 log(log x) 4) Let {an}^__x = {§ — arctann} _ with the associate function / (x ) I — arctan x. Then logx logx lim 7 ——- = lim l i l i i ——— - — 11111 ; z->oo - l 0 g ( - / ' ( j ; ) ) x-+°° ~ log y ^ - j = i i m l o g x = 1 :r->oo log( l + X2) 2 ' 370 CALCULATION OF METRIC DIMENSION OF CONVERGING SEQUENCES 5) Let {On}^° x = {l — e~™ }™=1 with the associate function f(x) — 1 — e * Calculate .. logx logx h m 7 - — T - = l i m x-̂ oo - log(-/'(x)) *->oo _ loJ<ri_\ logx lш logx 1 = lim гr = lim — x—>-oo l o g ( x 2 e i ) ^ ^ o o l + l o g x 2 2 Open questions We can hardly expect that the corresponding result for the lower dimen­sion holds. The open problem remains to investigate stronger relations among lO X I O S / ( . E ) liminf- -,—, liminf | o g ( ~ / ( z ) ) and dim A than are those formulated in ^ o o l o s - 7 ^ y X-*CK l o S 3 7 ^ ) Proposition 1. Another open problem is whether the reverse implication to that of Theorem 1 is true: If dim A exists, then also lim -— x exists. x^oo l o S f/(x) R E F E R E N C E S [BT] BESICOVITCH, A. S. TAYLOR, S. J. : On the complementary intervals of a linear closed sets of zero Lebesgue measure, J. London Math. Soc. 29 (1954), 449 459. [H] HAWKES, J . : Hausdorff measure, entropy and the independents of small sets, Proc. London Math. Soc. (3) 28 (1974), 700 724. [KA] KOQAK, §• AZCAN, H . : Fractal dimensions of some sequences of real numbers, Doga Mat. 17 (1993), 298 304. [KT] K O L M O G O R O V , A. N. T I K H O M I R O V , V. M. : s-entropy and e -capacity of sets in functional spaces, Uspekhi Mat. Nauk 14 (1959), 3 86 (Russian); In: Amer. Math. Soc. Transl. Ser. 2 Vol. 17, Amer. Math. S o c , Providence, RI, 1961, pp. 277 364. [MZ1] MISIK, L. ZACIK, T . : On some properties of the metric dimension, Comment . M a t h . Univ. Carolin. 3 1 (1990), 781 791. [MZ2] MISIK, L. ZACIK, T . : A formula for calculation of metric dimension of converging sequences, Comment. Math. Univ. Carolin. 4 0 (1999), 393 401. 371 TIBOR ŽÁČIK — LADISLAV MIŠÍK [PS] P O N T R Y A G I N , L. S — SNIRELMAN, L. G . : Sur une propriete metrique de la di-mensioщ Ann. of Math. (2) 3 3 (1932), 156-162 (Appendix to the Russian trans lation of Dimension Theory by W. Hurewitcz and H. Wa l lman, Izdat. Inostr. Lit., Moscow, 1948). Received December 15, 2003 Mathematical Institute Slovák Academy of Sciences Stefánikova 1^9 SK-814 13 Bratislava SLOVÁK REPUBLIC E-mail: zacik@mat.savba.sk c Department of Mathematics Faculty of Science University of Ostrava 30 dubna 22 CZ-101 03 Ostrava CZECH REPUBLIC E-mail: Ladislav.Misik@osu.cz 372

A simplified formula for calculation of metric dimension of converging sequences

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A simplified formula for calculation of metric dimension of converging sequences

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Institute of Mathematics AS CR, v. v. i.