On properties of periodically monotone sequences

NOIt'IH.~

On Properties of Periodically Monotone

Sequences

Giovanni Fiorito

Department of Mathematics Universit~z di Catania Viale Andrea Doria 6 Catania, Italy

Transmitted by Melvin Scott

ABSTRACT In this paper we introduce the periodically monotone sequences. Then we investigate the properties of a class of periodically monotone sequences in R. These sequences arise, for example, in counting as the integers are distributed on intervals of the form [(k - 1)T, kT[ where k ~ ~ and T ~ R +.

1.

INTRODUCTION

For every real n u m b e r x, as usual, [ x] is the greatest integer less t h a n or equal to x. Let T ~ ~+. If T>~ 1 we denote [ T ] by p and, V n ~ ( n >~ T), we denote by sn, T the n u m b e r of intervals [ ( k - 1)T, kT[ (k ~, kT ~ n) where there are p + 1 natural numbers, similarly, V n ~ N ( n >/ T), we denote by ~'n, T the n u m b e r of intervals [(k - 1)T, kT[ (k ~, k T < n) where there are p natural numbers (for n < T, we define

~ , ~ = E:~,r = 0). Likewise, if T < 1 we denote by p the greatest integer such t h a t 1

pT ~< 1 and, V n ~ ~ , by ¢rn, ~ the n u m b e r of intervals [k - 1, k[ (k ~ ~l,k

1We remark that p = [ 1 ] .

APPLIED MA THEMATICS AND COMPUTATION 72:259-275 (1995) © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010

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260

G. FIORITO

n) where there are p + 1 points of the form h T ( h ~ N); similarly, V n ~ N, we denote by ~r,~,V the n u m b e r of intervals [ k - 1, k{ (k ~ N, k ~< n) where there are p points of the form h T ( h E N). In the first section of this paper we introduce the periodically monotone sequences (a suitable generalization of arithmetical progressions and of periodical sequences) and prove some preliminary results. In the second section we give the a n a l y t i c expressions of s,~, T, e',~,T, (~,, T, tr,:, r" From these we deduce then, in some cases, the divergence (to + ~) of the sequences

we also prove t h a t these sequences are definitively periodically increasing if and only if T is rational and, in some cases, not integer. Furthermore, we prove the convergence of the sequences

and we found their limits. Finally we remark that if T is irrational, some of the results given above m a y also be deduced from the uniform distribution in [0, TI of the sequences {r n} where 2 % = n - [ n / T ] T V n ~ N. Likewise, other results m a y also be deduced from the uniform distribution in [0, 1] of the sequence { n T - [ nT]}. At the end of the second section we give some numerical examples t h a t emphasize the strange variation of these sequences. In all examples the evaluation of the terms of the sequences is done using a personal computer. In what follows, if T ~ ( ~ + - N, we set T = u / v and suppose t h a t u is prime to v. 2.

PERIODICALLY MONOTONE

SEQUENCES

Let { x n} be a sequence in N.

DEFINITION 2.1. such t h a t

{ X,~} is called periodic if there exists a natural n u m b e r q :cn+q=x ~

YneN;

(*)

the lowest natural n u m b e r q, for which ( * ) holds, is called period. 2In [1], using Weyl's theorem, we proved that if T ~ R + - Q then {%} is uniformly distributed in [0, TI.

Periodically Monotone Sequences

261

DEFINITION 2.2. { x n} is called periodically m o n o t o n e if there exist a natural n u m b e r q and a real n u m b e r k such t h a t 3

x,,+q=x,,+k

Vn~N.

(**)

In particular, if k > 0 (k < 0) { x n} is called periodically increasing (decreasing). The lowest natural n u m b e r q for which (* *) holds is called period.

THEOREM 2.1.

Let {x,~} be a sequence verifying (* *). Then it results

xn lira ,~-~ n

k q"

PROOF. T h e thesis follows easily observing t h a t for i = 1, 2 . . . . , q and Yh~ Nwehave

xhq +~

x~ + hk

hq+ i

hq+ i

REMARK 2.1. The previous definitions and Theorem 2.1 m a y be easily extended supposing t h a t { x,} is a sequence in a Hausdorff topological vector space. The following lemmas give two interesting examples of periodically m o n o t o n e sequences t h a t will be utilized in the next section.

LEMMA 2.1. If T ~ •+ then the sequence { [ n / T ] } is periodically increasing; if T ~ N the period is T, otherwise the period is u. If T ~ R + - Q then {[ n / T ] } is not definitively periodically monotone.

3If k = 0 then {x,~} is periodic; if k ¢ 0 and q = 1 then (x~} is an arithmetical progression.

262 PROOF.

G. FIORITO We observe that V n, r ~ t~ it results: + 3

,

(1)

where 6 ( n / T ) and 6((n + r ) / T ) denote the fractional parts of n~ T and ( n + r ) / T, respectively. Therefore, if there exist q, k, v ~ N such that

we have also:

=~

+¥-

From (2), by taking into account that the sequence {6 ( n / T ) } is bounded it follows q --

T

-

k =

0.

(3)

From (3) we deduce easily that T is rational; moreover, if T ~ ~, q is a multiple of T and, if T ~ Q + - ~, q is a multiple of u. On the other hand, if T ~ we have

Vn E ~ ,

and, if T ¢ Q + -

N nq-u

T]

Vn ~ ~1.

This completes the proof.

LEMMA 2.2. If T ~ ~+ then the sequence {~,} is periodic; if T ~ ~q the period is T, otherwise the period is u. If T ~ R + - Q then the sequence {~'n} is not definitively periodic but it is dense in [0, T].

Periodically Monotone Sequences

263

We observe t h a t V n ~ N we have

PROOF.

% = 8

(-;)

T,

(We are using the same notations of L e m m a 2.1). Therefore, if there exist n, r ~ N such t h a t T~+, = ~-~, we have

F r o m this relation and from (1) of L e m m a 2.1 it follows t h a t r

F r o m (4) it follows easily t h a t r / T is a natural n u m b e r and t h a t T is rational. Moreover, if T ~ N, r is a multiple of T and, T ~ • + - N, r is a multiple of u. On the other hand, if T ~ N, we have V n ~ I~1

and, if T ~ ¢ ~ + - I~, we have Vn~. Finally, by taking into account that, if T ~ R + - Q, the sequence { ~ ( n / T ) } is dense in [0, 1] for Kronecker's t h e o r e m (for its s t a t e m e n t see [2, p. 373]), it follows t h a t {T~} is dense in [0, T]. This completes the proof. • 3.

A CLASS OF P E R I O D I C A L L Y M O N O T O N E

THEOREM 3.1.

SEQUENCES

Let T ~ R + and T >~ 1. Then it results that

(1) O n - p

- [r~] - 1

ifT~ or

if T ~

ifT~

N

and and

n < T

n > T;

264

G. F I O R I T O

(2) lim

ifT~

+~

e~, T =

~;

n--~

(3)

e~, T lim n-~

T-

n

- T

p '

(4) If T ~ Q + - ~ then {Sn, ~} is definitively periodically increasing of period u. If T ~ R + - Q then {s~, T} is not definitively periodically monotone.

PROOF. L e t u s s u p p o s e T ~ ~ + - N (if T ~ M, t h e t h e s i s is o b v i o u s b e i n g ~ , T = 0 V n ~ ~ a n d p = T). V n ~ ~ let k~ b e t h e g r e a t e s t i n t e g e r s u c h t h a t k n T < n a n d let 7~ b e t h e n u m b e r of i n t e g e r s m s u c h t h a t k~ T < m ~< n. T h e r e f o r e V n ~ ~1 ( n > T ) it r e s u l t s 4

n = pk n + 8n, T + rn" From the above relation, by taking into account that r n = [T,~] + 1 it follows

1

k,, = [ n~ T], a n d

(5)

T h i s p r o v e s T h e o r e m 3.1(1). O n t h e o t h e r h a n d , b e i n g

4We remark that if T ~ Q + - N then in every interval of the form [(k - 1)T, kT[, where k - 1(~ ~l) is a multiple of v, there are p + 1 natural numbers. If T ~ R + - Q, then the existence of intervals of the form [(k - 1)T, kT[, where there are p + 1 natural numbers, may be proved, for example, by Kronecker's theorem. Indeed, fixed s ~]0, T - p[, for the abovementioned theorem, there exist h, k ~ N such that Ih - kTI < e. If h > kT, then it follows

kT
~
l)T;

therefore, dividing the interval [kT + s,(k + 1)T[ into p equal intervals, it follows that in every one of them (of length greater than 1) there is an integer. Likewise if h < kT.

265

Periodically Monotone Sequences

from (5) we deduce

By noticing, now, t h a t the sequences

are bounded, T h e o r e m 3.1(2) and (3) follow easily from (6). Now we prove (4) of T h e o r e m 3.1. W e observe at first t h a t Y n ~ ~ it results %+1 = % + 1

or

%+1 = % + 1 -

T

[%+11= [%] + 1

or

[%+11 = [ % - T] + 1.

and then

Therefore, if [%] = p, it results [%+1] = 0; if [%] = p - 1, it results [%+1] = p or [%+ 1] = 0; finally, if p > l a n d [ % ] < p 1, we h a v e % < p 1, % < T - 1, [% - T] < - 1 and therefore [%+ 1] -- [%] + 1. W e observe then t h a t V n, r ~ N ( n > p) it is

F r o m this relation and from (1) of L e m m a 2.1, we have

~,,+,-, r -- ~,~, r = r -

- ~ + p8

-- p8

-- [rn+~] + [r~].

Then, if there exist q, k, v ~ ~J such t h a t Sn+q, r = s,~,r + k

V n > v,

we have also V n > m a x ( v , p)

[%+ql

- p~[-T-

= [,n]

- p~

+ q -

-~

-

266

G. FIORITO

From (7), by taking into account that the sequence {[vn] - p S ( n / T ) } is bounded, it follows

Pq

q----

T

k=0.

(8)

From (8) we deduce easily that T is rational; moreover from (7) it follows [~n+q]-p~(u_~q)

--[Tn]-pS(T)

V n > max(e, p,.

(9,

Now, for the Lemma 2.2 we have

and so (9) holds also V n ~ N. On the other hand, we have

or

and therefore from (9) it follows

From (10), for n = p, by taking into account that [r v] = p, we deduce easily 8

q

Otherwise, if n = p + 1, by taking into account that [rp+ 1] = 0, from (10) again we deduce q ['rp+ l+q] = p B (-~)

or

[Tp+ l+q] = pS( T ) -

p.

(12)

Periodically Monotone Sequences

267

Now, the second relation of (12) is false; if ti( q~ T) > 0, the first relation of (11) is false and, for the initial remark in the proof of T h e o r e m 3.1(4), the second relation of (11) and the first of (12) are in contradiction, therefore 3 ( q / T ) = 0 and q is a multiple of u. Finally, we observe t h a t if T ~ Q + - ~l and n > p we have

=n+u-P[T]= ~,~

+ v(T

pv-

[7~] - 1

p);

-

the proof is completed.

THEOREM 3.2.

Let T ~ •+

and T >1 1. Then it results

(1) ifn<. T ~,

=

-1

ifTe~

and

n > T

ifT~

and

n > T;

n

+ [~.]

-

n + 1

(2) p

lim ~,~, T = + ~;

(3) ~r n,T

lim - n-~ n

=

p+l-T T

(4) If T ~ C~+ then {~'n,T} is definitively periodically increasing; if T ~ N the period is T, otherwise the period is u. If T ~ R + - Q then { ~'~,T} is not definitively periodically monotone.

G. FIORITO

268

PROOF.

If T •

Nand n>

T we have

if T ~ M and n > T it results

[-;]

gn, T -t- ~°n, T ~

and these relations imply easily (1)-(3) of Theorem 3.2. Now we prove (4) of Theorem 3.2. If T • N then (4) follows easily from Lemma 1.1. If T • N + t~, then Vn, r • N ( n > p) it results (p+

8:~+"' T -- S" r

T

1)r

(p+

+ ( p + 1)~

-~

(nTr)

1)6 - -

+ [ ~ , + , ] - [~,~] - ~.

Then, if there exist q, k, u • [~ such that !

¢

~ n + q , T ~ En, T-~- k

Vrt ~ P,

we have also V n > max(v, p) [7".+q]- ( p +

1)6

(')

1) 6 --~ + q

= [~',]- (p+ (p+l) T

q + k.

(13)

From (13), by taking into account that the sequence {[~',] - ( p + 1 ) 6 ( n / T ) } is bounded, it follows q

(p+

1)q T

+ k = 0.

(14)

From (14) we deduce easily that T is rational; moreover from (13) it follows [%~+q]- ( p +

1)¢3( n + q

Yn • N. (15)

269

Periodically Monotone Sequences

On the other hand we have

or

8(n-~q)

= 8{T)+

8{T)-1

and therefore from (15) it follows

[%+q]=[%]+(P+l)a(q)

or

[%÷~] = [%1 + (p + 1 ) ~ ( - ~ ) - p - 1 .

(16)

F r o m (16), if n = u - 1, by taking into account t h a t T,_I = u - 1 - ( v 1)T = T - 1 and hence [~u-1] = P - 1, we deduce easily

or

Otherwise, if n = u, by taking into account t h a t ~'u = [Tu] = 0, from (16) again we deduce

or

[Tu+q] = ( P - k

1)~(T

) - p -- 1. (18)

Now, the second relation of (18) is false; if (~( q~ T) > 0, the second relation of (17) and the first of (18) are in contradiction, and the same happens for the first of (17) and the first of (18); therefore 8 ( q / T ) = 0 and q is a multiple of u.

270

G. FIORITO

Finally, we observe t h a t if T ~ ~ } + - M and n > p we have

=(P+I)[T]

+(p+l)v+[%l-n-u+l

= s'~,T + v ( p + 1 -

T);

the proof is completed.

THEOREM 3.3.

Let T ~ ]0, 1[. Then it results

(1) ifT~ 1

-

-rip-1

i f ~ ;

(2) 1

lira O'n,T = +oo ~-.o

if--f f~ ~;

oo

(3) lim n-~

on, ~

1 - pT

n

T

'

(4) If 1 / T ~ Q + - t~ then {(rn, T} is periodically increasing of period u. If 1 / T ~ ~ + - C~ then {on, v} is not definitively periodically monotone.

PROOF. Let us suppose 1 / T ~ ~ (if 1 / T ~ ~, the thesis is obvious, being p T = 1 and try,T = 0 V n ~ ~). V n ~ ~ let k~ be the greatest integer such t h a t k n T < n. We have k ~ T < n < ~ ( k ~ + 1)T, k. < n i T

k~ + 1, - k ~ > - n / T > _ . - k s - 1, - k s - 1 = [ - n / T ] , k~ = - [ - n / T ] 1. Therefore V n ~ ~ it results k~ = n p + ¢r~,T; from this it follows

~o,== - [ - - ~ ] - rip-1.

(19)

271

Periodically Monotone Sequences

This proves (1) of Theorem 3.3. From (19) also it follows

n(1 ;)+ where 8 ( - n ~ T ) = - n~ T - [ - n / T ] , and from this relation we may deduce Theorem 3.3(2) and (3) immediately. Now we prove (4) of Theorem 3.3. We observe that V n, r ~ N we have

(r"+ r' r - ~rn'T

T

rp + ~

T

ti

.

Therefore, if there exist q, k, 1, ~ N such that O'n+q,T ~- O'n,T-~- k

V n > /.-',

we have

(

q

T

- -T + qp"

(20)

From (20), by taking into account that the sequence { 8 ( - n~ T)} is bounded, it follows q k+ qp-

(21)

-- -O.

T

From (21) we deduce easily that T is rational and that q is a multiple of u. On the other hand, if 1 / T ~ ( ~ + - ~ , we have V n ~ t~

n+u] O'n+u.T = --

= -I-T]

~

-- ( n +

+v-np-

u) p - - 1

1-vTp

=(r~,T+ v(1--pT). This completes the proof.

•

272

G. FIORITO

Let T ~ ]0, 1[. Then it results

THEOREM 3.4. (1)

{:

1

-1

,T

if-~

[

~

p+l)n+

1

-

+1

if--~N;

(2) lira o-" v =

+oo;

n ---) oc

(3) o',~,v lira n--*~

(p+l)

T-1

= n

T

(4) If 1/ T ~ Q+ then {~r~,v} is periodically increasing; if 1/ T ~ 3~ the period is 1, otherwise the period is u. If 1 / T ~ gO+- Q then {(r" T} is not definitively periodically monotone.

PROOF. If 1 / T ~ ~ the thesis is obvious. If 1 / T ~ ~, then to prove (1)-(3) of Theorem 3.4 it is sufficient to observe that Vn ~ ~ we have ~,~,T + ~,:,T = n.

(22)

Now we prove (4) of Theorem 3.4. If 1 / T ~ ~+ - ~ then we have Vn, r ~ O " ~ + " ' r - - O " ~ ' r = r ( P + 1)

T +8

~

T

Therefore, if there exist q, k, ~, ~ N such that O'~+q,r = O'~,T + k

V n > ],,,

we have

8

(

n+ q)= T

8(_T)

+ q(p+l)

q --T-

k.

(23)

Periodically Monotone Sequences

273

From (23), by taking into account that the sequence {8( - n~ T)} is bounded, it follows q(P+

q 1) - - - T

k=0.

(24)

From (24) we deduce easily that T is rational and that q is a multiple of u. On the other hand, if 1 / T ~ ~D+ - IN, we have Vn ~ IN nq-u

< + " 7 ' = ( n + u ) ( v + 1) +

T

+ 1 = •,T + v ( ( p + 1) T -

1).

This completes the proof.

RENIARK 3.1. If T ~ Q + - IN, (3) of Theorem 3.1 also follows from (4) of the same Theorem and from Theorem 2.1. A similar remark may be made for (3) of Theorems 3.2, 3.3, and 3.4.

REMARK 3.2. If T ~ ~ + - Q and T > 1, (3) of the Theorem 3.1 follows from Theorem 1.1 of [1] (on the uniform distribution of the sequence {%}) without using analytic expression of s,~, T, but only noticing that, in this case, s,~, T is equal to the numbers of terms of the sequence {%} obtained for k < [ n / T ] T and belonging to [0, T - p[. If T ~ ~ + - Q and T < 1, (3) of the Theorem 3.3 follows from Weyl's Theorem (on the uniform distribution of the sequence {kT - [ kT]}, see, [3]) without using analytic expression of ~rT~' r, but only noticing that, in this case, ~rn,T is equal to the number of terms of the sequence { k T - [ k T ] } obtained for k <~ [ n / T ] and belonging to [ pT, 1[.

EXAMPLE 1. Let T = 2.71. Then some terms of the sequences {s., r} and {d,,, r} are given by the formulas

=

i En, T

2618 /~ 8 p

n,T

100 I, 1070

forl~
274

G. FIORITO

For Theorems 3.1 and 3.2, the following recursive formulas 5 hold: En+ 271, T

°°n, T "~ 71

~'n+271, T

n, T + 29

Vn ~ ~.

(25)

EXAMPLE 2. Let T = 3.14. Then some terms of the sequences {e~, T} and {~,, T} are given by the formulas f o r l ~< n~<25 for 26~< n < 4 7 for 48~< n~<69 for 990 ~< n ~< 1011 for 9983 < n ~< 10004

44 445 i t 8n, T

forl~
= [2738

For Theorems 3.1 and 3.2, the following recursive formulas hold: ~tn+157, v ~

~n+157, T ~--- ~ n , T + 7

i 6~,r + 43

Vn ~ ~1.

(26)

EXAMPLE 3. Let T = 0.018. Then some terms of the sequence {(rn, T} are given by the formula i ~ r

=

[5555

for n = 1 for2~
For Theorem 3.3, the following recursive formula holds: O'n+9, T = O'n,T"[- 5

Vn E ~.

(27)

5We have checked, by a personal computer, the correctness of formulas (25), (26), and (27) up to n = 10 4.

Periodically Monotone Sequences

275

The author is thankful to Professor F. Guglielmino for the useful discussions he had with him. REFERENCES 1 G. Fiorito, R. Musmeci, and M. Strano, Uniforme distribuzione ed applicazioni ad una classe di serie ricorrenti, Le Matematiche 48:123-133 (1993). 2 G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford 1954. 3 L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York 1974.