CANTOR SERIES EXPANSIONS OF RATIONAL NUMBERS 1 ...

CANTOR SERIES EXPANSIONS OF RATIONAL NUMBERS SYMON SERBENYUK

Abstract. This article is devoted to necessary and sufficient conditions for a rational number to be representable by a positive Cantor series. Well-known results are described. Necessary and sufficient conditions are formulated for the case of an arbitrary sequence (qk ) and some them corollaries are considered.

1. Introduction Series of the form ε1 ε2 εk + + ··· + + ... q1 q1 q2 q1 q2 . . . qk

(1)

were considered by Georg Cantor in [1]. Here Q = (qk ) is a fixed sequence of positive integers, qk > 1, and (Θk ) is a sequence of the sets such that Θk = {0, 1, . . . , qk −1}, and εk ∈ Θk . In the present article, we use also the following notations: N, Z0 , Z, Q, and I. Here by N denote the set of all positive integers and by Z0 denote the set N ∪ {0}, Z is the set of all integers, and Q is the set of all rational numbers, and I is the set of all irrational numbers. In the paper [1], necessary and sufficient conditions for a rational number to be representable by series (1) are formulated for the case when (qk ) is a periodic sequence. The problem of expansions of rational or irrational numbers by Cantor series have been studied by a number of researchers. For example, P. A. Diananda, A. Oppenheim, P. Erd¨ os, J. Hanˇcl, E. G. Straus, P. Rucki, R. Tijdeman, P. Kuhapatanakul, V. Laohakosol, B. Mance, Pingzhi Yuan and others studied this problem. See [1]–[22] and [25]. Necessary and sufficient conditions for a rational number to be representable by a Cantor series is the main problem of the present article. Much research [25, 7, 10, 8, 11] has been devoted to necessary or/and sufficient conditions for a rational number to be representable by Cantor series (1) such that sequences (qk ) and (εk ) are sequences of integers. In some papers (see [10, 25, 5, 8, 15]), the case of Cantor series for which sequences (qk ) and (εk ) are sequences of integers and the condition Z 3 qk > 1 holds for all k ∈ N is investigated. However the main problem of the present article is studied for the case of series (1) (see [1, 2, 16, 18]) and still for the case of Cantor series of special type (see 2010 Mathematics Subject Classification. 11K55, 11J72. Key words and phrases. Cantor series, rational numbers, shift operator. 1

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[12, 14, 13, 10]). For example, in the papers [2, 11, 4], Ahmes series are considered. The last series is Cantor series (1) for which εk = const = 1 holds for all k ∈ N. In the papers [2, 7, 8, 10, 11, 12, 5, 18, 25], necessary and sufficient conditions for a rational (irrational) number to be representable by Cantor series are studied, and sufficient conditions are investigated in the papers [5, 2, 10, 16, 18, 25]. Although much research has been devoted to the problem of representation of rational (irrational) numbers by a Cantor series for which sequences (qk ) and (εk ) are sequences of special types (see [1, 5, 7, 8, 10, 16, 18, 25]), little is known about necessary and sufficient conditions of the rationality (irrationality) for the case of an arbitrary sequence (qk ) (see [2, 10, 25]). In the following section, let us consider the conference paper [21] published into Ukrainian (the last paper is a conference paper of the International Conference on Algebra dedicated to 100th anniversary of S. M. Chernikov, Kyiv, 2012). The main results of the last paper were published into English as the preprint [22]. To solve the main problem of the papers [21, 22], the notion of the shift operator is used. 2. On some results for the general case 2.1. Definitions. We begin with definitions. By x = ∆Q ε1 ε2 ...εk ... denote a number x ∈ [0, 1] represented by series (1). This notation is called the representation of x by Cantor series (1). Define the shift operator σ of expansion (1) by the rule ∞ X σ(x) = σ ∆Q ε1 ε2 ...εk ... = k=2

εk = q1 ∆Q 0ε2 ...εk ... . q2 q3 . . . qk

It is easy to see that σ n (x) = σ n ∆Q ε1 ε2 ...εk ... =

∞ X

εk

= q1 . . . qn ∆Q 0| .{z . . 0} εn+1 εn+2 ... . qn+1 qn+2 . . . qk k=n+1

(2)

n

Therefore, x=

n X i=1

εi 1 + σ n (x). q1 q2 . . . qi q1 q2 . . . qn

(3)

In the paper [23], the notion of the shift operator of an alternating Cantor series is studied in detail. 2.2. Rational numbers that have two different representations. Certain numbers from [0, 1] have two different representations by Cantor series (1), i.e., Q ∆Q ε1 ε2 ...εm−1 εm 000... = ∆ε1 ε2 ...εm−1 [εm −1][qm+1 −1][qm+2 −1]... =

m X i=1

εi . q1 q2 . . . qi

CANTOR SERIES EXPANSIONS OF RATIONAL NUMBERS

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Theorem 1. A rational number pr ∈ (0, 1) has two different representations if and only if there exists a number n0 such that q1 q2 . . . qn0 ≡ 0

(mod r).

Proof. Necessity. Suppose p ε1 εn−1 εn = + ··· + + r q1 q1 q2 . . . qn−1 q1 q2 . . . qn ε1 q2 . . . qn + ε2 q3 . . . qn + · · · + εn−1 qn + εn . = q1 q2 . . . qn Whence, εn =

pq1 q2 . . . qn − r(ε1 q2 . . . qn + ε2 q3 . . . qn + · · · + εn−1 qn ) . r

Since the conditions εn ∈ N and (p, r) = 1 hold, we obtain q1 q2 ...qn ≡ 0 (mod r). Sufficiency. Assume that (p, r) = 1, p < r, and there exists a number n0 such that q1 q2 . . . qn0 ≡ 0 (mod r). Then n

0 X ε1 ε2 εn0 εn0 +1 εi p = + + ··· + + + t n0 , + ··· = r q1 q1 q2 q1 q2 . . . qn0 q1 q2 . . . qn0 qn0 +1 q q . . . qi 1 2 i=1

where tn0 is the residual of series, and ε1 q2 . . . qn0 + ε2 q3 . . . qn0 + · · · + εn0 p = + tn0 . r q1 q2 . . . qn0 Clearly, pθ = (ε1 q2 . . . qn0 + ε2 q3 . . . qn0 + · · · + εn0 ) + q1 q2 . . . qn0 tn0 , q q ...q

where θ = 1 2 r n0 . Since pθ is a positive integer number, we have q1 q2 . . . qn0 tn0 = 0 or q1 q2 . . . qn0 tn0 = 1. Q That is x = ∆Q ε1 ε2 ...εn0 −1 εn0 000... in the first case and x = ∆ε1 ε2 ...εn0 −1 [εn0 −1][qn0 +1 −1][qn0 +2 −1]... in the second case. 2.3. The main theorems about representations of rational numbers. Theorem 2. A number x represented by series (1) is rational if and only if there exist numbers n ∈ Z0 and m ∈ N such that σ n (x) = σ n+m (x). Proof. Necessity. Let we have a rational number x = uv , where u < v and (u, v) = 1. Consider the sequence (σ n (x)) generated by the shift operator of expansion (1) of

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the number x. That is σ 0 (x) = x, σ(x) = q1 x − ε1 , 2

σ (x) = q2 σ(x) − ε2 = q1 q2 x − q2 ε1 − ε2 , .....................   n n−1 Y X σ n (x) = qn σ n−1 (x) − εn = x qi −  εj qj+1 qj+2 . . . qn  − εn , i=1

j=1

..................... From equality (3) it follows that un uq1 q2 . . . qn − v(ε1 q2 . . . qn + · · · + εn−1 qn + εn ) = . (4) v v Since v = const and the condition 0 < uvn < 1 holds as n → ∞, we see that un ∈ {0, 1, . . . , v − 1}. Thus there exists a number m ∈ N such that un = un+m . In addition, there exists a sequence (nk ) of positive integers such that unk = const for all k ∈ N. Sufficiency. Suppose there exist n ∈ Z0 and m ∈ N such that σ n (x) = σ n+m (x). In our case, from equality (2) it follows that qn+1 . . . qn+m ∆Q . . 0 ε ε ...ε . x = ∆Q ε1 ε2 ...εn 000... + qn+1 . . . qn+m − 1 0| .{z } n+1 n+2 n+m 000... σ n (x) =

n

Thus one can formulate the following proposition. Lemma 1. If there exist numbers n ∈ Z0 and m ∈ N such that σ n (x) = σ n+m (x), then x is rational and the following equality holds: q1 q2 . . . qn qn+1 . . . qn+m Q σ n (x) = ∆0 . . . 0 ε ε ...ε . qn+1 . . . qn+m − 1 | {z } n+1 n+2 n+m 000... n

This completes the proof. Theorem 2 proved.

Theorem 3. A number x = ∆Q ε1 ε2 ...εk ... is rational if and only if there exist numbers n ∈ Z0 and m ∈ N such that Q ∆Q 0 . . 0} εn+1 εn+2 ... = qn+1 . . . qn+m ∆0| .{z . . 0} εn+m+1 εn+m+2 ... . | .{z n

n+m

Theorem 2 and Theorem 3 are equivalent. 2.4. Certain corollaries of the main theorem. Consider the condition σ n (x) = const for all n ∈ Z0 . It is easy to see that there exist numbers x ∈ (0, 1) such that the last-mentioned condition is true, e.g. 1 2 3 n 1 x= + + + ··· + + · · · = σ n (x) = . 3 3·5 3·5·7 3 · 5 · . . . · (2n + 1) 2


Lemma 2. If σ n (x) = x for all n ∈ N, then

εn qn −1

5

= const = x.

Proof. If σ n (x) = σ n+1 (x) = const, then σ n (x) = qn+1 σ n (x) − εn+1 . Whence, εn+1 σ n (x) = = const. (5) qn+1 − 1 That is x=

n−1 X ε1 ε1 ε2 εn εi = + = ··· = + = .... q1 − 1 q1 q1 (q2 − 1) q q . . . q q q . . . q i 1 2 n−1 (qn − 1) i=1 1 2

Remark 1. If the condition σ n (x) = const holds for all n ≥ n0 , where n0 is a fixed n positive integer number, then from (3) it follows that the condition qnε−1 = const holds for all n > n0 . Lemma 3. Let n0 be a fixed positive integer number. Then the condition σ n (x) = n const holds for all n ≥ n0 if and only if qnε−1 = const for all n > n0 . Proof. Necessity follows from the previous lemma. Sufficiency. Suppose that the condition εn+1 εn+i εn = = ··· = = ... const = p = qn − 1 qn+1 − 1 qn+i − 1 holds for all n > n0 . Using the equality

εn qn

=

εn qn −1

−

εn qn (qn −1) ,

we get

∞ X

εi εn+1 εn+1 σ (x) = − = q . . . qi qn+1 − 1 qn+1 (qn+1 − 1) i=n+1 n+1   n+i ∞ X Y εn+i+1 εn+i+1 1  + − q − 1 q (q − 1) q n+i+1 n+i+1 n+i+1 i=1 j=n+1 i   n+i ∞ X Y 1 1 1  1− =p 1− +p = p. qn+1 q − 1 q n+i+1 i=1 j=n+1 j n

It is easy to see that the following statement is true. Proposition 1. The set {x : σ n (x) = x ∀n ∈ N} is a finite set of order q = minn qn ε and x = q−1 , where ε ∈ {0, 1, . . . , q − 1}. Lemma 4. Suppose we have q = minn qn and a fixed number ε ∈ {0, 1, . . . , q − 1}. ε n −1 Then the condition σ n (x) = x = q−1 holds if and only if the condition qq−1 ε = εn ∈ Z0 holds for all n ∈ N.

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Proof. Necessity follows from Proposition 1 and equality (5). n −1 ε. Then Sufficiency. Suppose εn = qq−1 x=

∞ X

qn −1 q−1 ε

n=1

q1 q2 . . . qn

=

∞ ε X qn − 1 ε = . q − 1 n=1 q1 q2 . . . qn q−1

Corollary 1. Let n0 be a fixed positive integer number, q0 = minn>n0 qn , and ε0 ε in expansion (1) of x providing be a numerator of the fraction q1 q2 ...qn qnn0 +k 0 0 +1 ...qn0 +k n that qn0 +k = q0 . Then σ (x) = const for all n ≥ n0 if and only if the condition qn −1 q0 −1 ε0 = εn ∈ Z0 holds for any n > n0 . Let us consider relationship (4). The main attention will be given to the existence of the sequence (nk ) such that unk = const for all k ∈ N in (4). Let we have a rational number x ∈ [0, 1]. Then there exists the sequence (nk ) such that σ nk (x) = const. Hence we get const =

=

∞ X εi εi = = ... q . . . qi q . . . qi i=n +1 n2 +1 +1 n1 +1

∞ X

i=n1 ∞ X i=nk

2

εi = .... q . . . qi +1 nk +1

It is easy to see that Cantor series (1) for which the condition σ nk (x) = const holds can be written by a certain Cantor series for which the condition σ k (x) = const holds. That is n1 ∞ X X 0 εj εk 1 x= = + x, q1 q2 . . . qk q q . . . q q q . . . q j 1 2 n1 j=1 1 2 k=1

0

x =

∞ X k=1

εnk +1 qnk +2 qnk +3 . . . qnk+1 + εnk +2 qnk +3 . . . qnk+1 + εnk+1 −1 qnk+1 + εnk+1 (qn1 +1 . . . qn2 )(qn2 +1 . . . qn3 ) . . . (qnk +1 . . . qnk+1 ) =

∞ X k=1

λk . (µ1 + 1) . . . (µk + 1)

(6)

0

In the case of series (6), the condition σ k (x ) = const holds for all k = 0, 1, . . . . Clearly, the following statement is true. Theorem 4. A number x represented by expansion (1) is rational if and only if there exists a subsequence (nk ) of positive integers such that for all k = 1, 2, . . . , the following conditions are true: • εn +1 qnk +2 . . . qnk+1 + εnk +2 qnk +3 . . . qnk+1 + · · · + εnk+1 −1 qnk+1 + εnk+1 λk = k = const; µk qnk +1 qnk +2 . . . qnk+1 − 1


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• λk = µµk λ, where µ = mink∈N µk and λ is a number in the numerator of the fraction whose denominator equals (µ1 + 1)(µ2 + 1) . . . (µ + 1) from sum (6). Let x be a rational number. Then we have ∞ X εk x= q1 q2 . . . qk k=1

=

n X i=1

=

n X i=1

X ∞ εi εn+1 εn+m εn+m+j + + ··· + + q1 q2 . . . qi q1 q2 . . . qn+1 q1 q2 . . . qn qn+1 . . . qn+m q q . . . qn+m+j j=1 1 2

∞ εi εn+1 qn+2 . . . qn+m + · · · + εn+m−1 qn+m + εn+m X εn+m+j + + . q1 q2 . . . qi q1 q2 . . . qn (qn+1 . . . qn+m ) q q . . . qn+m+j 1 2 j=1

From Theorem 2 it follows that εn+1 qn+2 ...qn+m + ... + εn+m−1 qn+m + εn+m . σ n (x) = σ n+m (x) = qn+1 ...qn+m − 1 Hence the following statement is true. Theorem 5. If a number x represented by Cantor series (1) is a rational number (x = uv ), then there exist n ∈ Z0 and m ∈ N such that q1 q2 . . . qn (qn+1 qn+2 . . . qn+m − 1) ≡ 0

(mod v).

3. Other results Consider other papers. Now we note the most interesting results about representation of rational or irrational numbers by Cantor series. The following results are somewhat similar to the corollaries of the main theorem in [21, 22]. In [2], the following statement that is similar to Theorem 4 was formulated by P. H. Diananda and A. Oppenheim. The authors considered Cantor series of form (1) in this paper. Theorem 6 ([2]). A necessary and sufficient condition that x given by (1) shall be rational is this: coprime integers h, k, 0 ≤ h ≤ k, an integer N and a condensation shall exist such that h Ai = (Bi − 1) k for all i ≥ N . Here

A1 A2 An + + ··· + + ..., B1 B1 B2 B1 B2 · · · Bn where A0 = ε0 is the integer part of x, B1 = q1 q2 · · · qi1 , B2 = qi1 +1 qi2 +1 · · · qi1 +i2 , . . . , and Bi ≥ 2, 0 ≤ Ai ≤ Bi − 1, ε1 ε2 εi1 A1 + + ··· + = . q1 q1 q2 q1 q2 · · · qi1 B1 x = X = A0 +

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For example, consider the product B2 = qi1 +1 qi2 +1 · · · qi1 +i2 . From the number of multipliers in this product, it follows that, in [2], a sequence other than the sequence (in ) (for the last sequence the condition ϕˆin (x) = const holds in the proof of Theorem 4) is used. Moreover, all possible values of the constant hk are indicated in Theorem 4. The proof of Theorem 6 in [2] and the proof of Theorem 4 in [21, 22] are different. In [2], the interest of Theorem 6 is indicated since this statement includes conditions of the rationality given by Cantor for the following two cases in [1]: for the following cases: • when a sequence (qk ) is a sequence such that for any r ∈ Z the condition . q q · · · q .. r holds for all large k; 1 2

k

• when a sequence (qk ) is periodic. In the first case, we obtain the result formulated in Theorem 1. The sufficient condition of this theorem was proved by Cantor in [1] but this statement had the other formulation and the more complicated proof. In the second case, x is a rational number if and only if (qk ) is ultimately periodic. The last statement was completely proved by Cantor in [1]. Let us consider results related with Theorem 1. In 2006, J. Sondow gave a geometric proof of the irrationality of the number e [24]. In [17], the following statement was proved by a generalization to Sondows construction. P∞ Theorem 7 ([17]). Let x = k=1 q1 qε2n···qk . Suppose that each prime divides infinitely many of the qk . Then x ∈ I if and only if both 0 < εk < qk − 1 hold infinitely often. The following statement is interesting as well. P∞ εk r Lemma 5 ([10]). If S = k=1 q1 q2 ···qk = p for some r ∈ Z and p ∈ N, then pSN ∈ Z for all N ∈ N, where SN =

∞ X n=N

εn . qN · · · qn

Remark 2. It is easy to see that there exist sequences (qk ) and (εk ) such that a finite expansion is a necessary or/and sufficient condition of the rationality of any number represented by a Cantor series. Several papers were devoted to these investigations. For example, see [16, 8]. The main results of the paper [8] are the following statements. Theorem 8 ([8]). Let (qk ) be a sequence of positive integers greater than one. Suppose that (εk ) is a sequence of integers such that lim inf

k→∞

|εk | + 1 =0 qk


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and for every sufficiently large positive integer k 1 |εk+1 | ≤ max (|εk |, 1)qk+1 . 2 P∞ Then n=1 q1 qε2k···qk ∈ Q if and only if εk = 0 for every sufficiently large positive integer k. Theorem 9 ([8]). Let (qk ) be a sequence of positive integers greater than one, and K ∈ (0, 1). Suppose that (εk ) is a sequence of non-negative integers such that lim inf

n→∞

εk + 1 =0 qk

and for every sufficiently large positive integer k εn+1 ≤ K max (εn , 1)dn+1 . P∞

εk k=1 q1 q2 ···qk

Then integer k.

∈ Q if and only if εk = 0 for every sufficiently large positive

Now we consider results related with a statement and a proof of Theorem 4. That k = const (the last is consider several results about cases when the condition qkε−1 equality holds for all k greater than some fixed k0 ) is a necessary and/or sufficient condition for a rational number to be representable by a Cantor series. See [2, 10, 25]. In [10], J. Hanˇcl and R. Tijdeman formulated conditions of the irrationality of a number represented by Cantor series (1) when sequences (qk ) and (εk ) are sequences of positive integers and qk > 1 for all k ∈ N. They used a notion of the following sum in the last-mentioned article: ∞ X εn SN = . qN · · · qn n=N

Note that, in [10], the authors noted that sum (1) is equal to a rational number if εk qk −1 = const holds for all k greater than some number n0 . This article is partially k devoted to conditions under which the condition qkε−1 = const is a necessary and sufficient condition of the rationality of a number represented (1). In by expansion particular the following cases are considered: lim inf n→∞

εn+1 qn+1

−

εn qn

= 0, εn =

o(qn−1 qn ), εn+1 − εn = o(qn−1 qn ). We note the following results. Proposition 2 ([10]). If (Sn ) is boundedPfrom below and for every ε > 0 we have ∞ n Sn+1 − Sn < ε for n ≥ n0 (ε), then S = n=1 q1 qε2n···qn ∈ Q if and only if qnε−1 = const for N > N0 . CorollaryP 2 ([10]). If (εn ) is a sequence of positive integers such that εn+1 − εn = ∞ εn = const for n greater than some n1 . o(n), then n=1 εn!n ∈ Q if and only if n−1 Theorem 10 ([10]). P∞Let (qn ) be a monotonic sequencenof positive integers satisfying εk = o(qn2 ). Then n=1 q1 qε2n···qn ∈ Q if and only if qnε−1 = const for n ≥ n0 .

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Theorem 11P([10]). Let (qn ) be an unbounded monotonic sequence of positive in∞ tegers. Then n=1 q1 q2n···qn ∈ Q if and only if dnn−1 = const for n ≥ n0 . Results obtained in [10] were generalized and corrected by Robert Tijdeman and Pingzhi Yuan in the paper [25]. In particular results is generalized for the cases when εn = n and qn → ∞, qn = n and εn+1 − εn = O(n). In the last-mentioned n = const for all n ≥ n0 be paper, it is shown that, in order that the condition qnε−1 a necessary and sufficient condition of the rationality, one can neglect the condition εn = o(qn2 ) in the system of conditions: εn = o(qn2 ) , εn ≥ 0, εn+1 − εn < εqn for n ≥ n1 (ε). We note the following statements. Theorem 12 ([25]). Let (qn ) be a monotonic integer sequence with qn > 1 for all such that εn+1 − εn = o(qn+1 ). Then P∞n andεn (εn ) be an integer sequence εn ∈ Q if and only if = const for all n greater than some n0 . n=1 q1 q2 ···qn qn −1 P∞ In this paper, other statements about conditions under which n=1 q1 qε2n···qn ∈ Q n = const for all n greater than some n0 are formulated. In if and only if dnε−1 addition, the following sufficient condition of the irrationality is proved. Theorem 13 ([25]). Suppose P∞ that qn > 1 for all n, that εn = O(qn ) and that limn→∞ εqnn = α ∈ I. Then n=1 q1 qε2n···qn ∈ I. The last statement with the condition 0 ≤ εn < dn without εn = O(qn ) was proved in [18]. In [25], the following denotations are used in proofs: S=

∞ X k=1

k ∞ X X ε∗j ε∗j ε∗k , S , R . = = nk nk ∗ ∗ q1∗ q2∗ · · · qk∗ q ∗ q ∗ · · · qj∗ qk+1 qk+2 · · · qj∗ j=1 1 2 j=k+1

Here (nk ) is a subsequence of positive integers, n0 = 1, ε∗k = εnk −1 + εnk −2 qnk −1 + · · · + εnk−1 qnk −1 qnk −2 · · · qnk−1 +1 , and d∗k = qnk −1 qnk −2 · · · qnk−1 , k = 1, 2, 3, . . . . For series (1), where (dn ) and (εn ) are sequences of integers such that qn > 0 for all n ∈ N and series (1) converges, the following statements are true. Lemma 6 ([25]). If there exists a subsequence (nk ) of positive integers such that Rnk = Rnk+1 for k = 1, 2, . . . , then S ∈ Q. Proposition 3 ([25] ). If (Rn ) is bounded from below and there exists a subsequence (nk ) of positive integers with Rnk+1 − Rnk < ε for k ≥ k0 (ε), then S ∈ Q if and only if Rnk = Rnk+1 for all large k. In [18], A. Oppenheim studied sufficient conditions of the irrationality of numbers represented by Cantor series (1) and also alternating series (1) such that |εi | < qi −1 for i = 1, 2, 3, . . . , and εm εn < 0 for some m > i and n > i when i is any fixed integer. In addition, in 2013 (see the presentation (in Ukrainian) and the working paper (in Ukrainian) that available at https://www.researchgate.net/publication/303720347,


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https://www.researchgate.net/publication/316787375 respectively), an expansion of numbers by an alternating Cantor series ∞ X (−1)k εk , εk ∈ {0, 1, . . . , qk − 1}, 1 < qk ∈ N, q1 q2 · · · qk

k=1

was investigated as a numeral system in the paper [23]. Necessary and sufficient conditions of the rationality of numbers represented by the last series are formulated in the last-mentioned article. These statements are proved in the last-mentioned presentation and the working paper as well. Remark 3. The technique of proving conditions of the rationality of numbers, considered in [21, 22] can be used for investigation of the general case when the last series is a sign-variable series. These investigations will be discussed by the author of the present article in a further paper. Also, in [18], the main results obtained by using some results from [1] and sums of the form εin +1 εin +2 εi + + ..., x in = n + qin qin qin +1 qin qin +1 qin +2 where (in ) is some subsequence of positive integers, and by investigation of a limit ε of cin = diin as n → ∞. n

Lemma 7. [18] A necessary and sufficient condition that x given by the convergent series (1), where qn and εn are integers, shall be irrational is that for every integer p ∈ N we can find an integer r ∈ Z and a subsequence (in ) such that r r+1 < xin < , n = 1, 2, 3, . . . . p p Necessary and sufficient conditions for a rational number to be representable by Cantor series of a special form are investigated by P. Erd¨os and E. G. Straus in [5]. Theorem 14 ([5]). Let (εn ) be a sequence of integers and (qn ) be a sequence of positive integers with qn > 1 for all large n and lim

n→∞

|εn | = 0. qn−1 qn

P∞ Then n=1 q1 qε2n···qn ∈ Q if and only if there exist a positive integer B and a sequence of integers (cn ) such that for all large n we have qn Bεn = cn qn − cn+1 , |cn+1 | < . 2 Theorem 15 ([5]). Let pn be the nth prime and let (qn ) be a monotonic sequence of positive integers satisfying pn qn lim 2 = 0, lim inf = 0. n→∞ qn n→∞ pn P∞ Then n=1 q1 qp2 n···qn ∈ I.

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Note also that, in [15], Bill Mance considered conditions of the irrationality of numbers, that have a property of a certain type of the normality and represented by Cantor series (1) for which εn 6= qn − 1 infinitely often. For example, the following result is interesting. Definition ([15, p.45]). A number x ∈ [0, 1) is called Q-distribution normal if the sequence X = (x

(mod 1), q1 x

(mod 1), q1 q2 x (mod 1), q1 q2 · · · qn x (mod 1), . . . )

is uniformly distributed in [0, 1). Theorem 16 ([15, p. 264]). A number x ∈ [0, 1) is irrational if and only if there exists a basic sequence Q = (qn ) such that x is Q-distribution normal. Finally, several papers (see [16, 9, 25]) were devoted to investigations of conditions P of the rationality or the irrationality of numbers represented by series of the ∞ form n=1 abnn . Furthermore, in [16], a necessary and sufficient condition of the n+1 P∞ rationality of the sum n=1 an (−1) is proved for the case of certain properties, bn that sequences (an ) and (bn ) satisfy them. References [1] Cantor G. Ueber die einfachen Zahlensysteme. Z. Math. Phys. — 1869. — Bd. 14. — S. 121– 128. [2] Diananda P. A., Oppenheim A. Criteria for irrationality of certain classes of numbers II. Amer. Math. Monthly. — 1955. — 62, No. 4. — P. 222-225. P 1 [3] Erd¨ os P., R´ enyi A. On Cantor’s series with convergent . Ann. Univ. Sci. Budapest E¨ otv¨ os qn Sect. Math. — 1959. — 2. — P. 93109. [4] Erd¨ os P., Straus E. G. On the irrationality of certain Ahmes series. J. Indian. Math. Soc. — 1968. — 27. — P. 129-133. [5] Erd¨ os P., Straus E. G. On the irrationality of certain series. Pacific J. Math. — 1974. — 55, No. 1. — P. 8592. [6] Galambos J. Representations of real numbers by infinite series. Lecture Notes in Math., Springer-Verlag, Berlin, Hiedelberg, New York. — 1976. — 502. [7] Hanˇ cl J. A note to the rationality of infinite series I. Acta Math. Inf. Univ. Ostr. — 1997. — 5, No. 1. — P. 5-11. ˇ at concerning series of Cantor type. Acta [8] Hanˇ cl J. A note on a paper of Oppenheim and Sal´ Math Inf. Univ. Ostr. — 2002. — 10, No. 1. — P. 3541. [9] Hanˇ cl J., Rucki P. A note to the transcendence of special infinite series. Mathematica Slovaka. — 2006. — 56, No. 4. — P. 409-414. [10] Hanˇ cl J., Tijdeman R. On the irrationality of Cantor series. J. Reine Angew. Math. — 2004. — 571. — P. 145158. [11] Hanˇ cl J., Tijdeman R. On the irrationality of Cantor and Ahmes series. Publ. Math. Debrecen. — 2004. — 65, No. 3-4. — P. 371380. [12] Hanˇ cl J., Tijdeman R. On the irrationality of factorial series. Acta Arith. — 2005. — 118. — P. 383-401. [13] Hanˇ cl J., Tijdeman R. On the irrationality of factorial series III. Indag. Mathem. — 2009. — 20, No. 4. — P. 537-549. [14] Hanˇ cl J., Tijdeman R. On the irrationality of factorial series II. J. Number Theory. — 2010. — 130, No. 3. — P. 595607. [15] Mance B. Normal Numbers with Respect to the Cantor Series Expansion. Dissertation, The Ohio State University, 2010.


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[16] Kuhapatanakul P., Laohakosol V. Irrationality of some series with rational terms. Kasetsart J. (Nat. Sci.). — 2001. — 35. — P. 205-209. [17] Marques D. A geometric proof to Cantor’s theorem and an irrationality measure for some Cantor’s series. Annales Mathematicae et Informaticae. — 2009. — 36. — P. 117-121. [18] Oppenheim A. Criteria for irrationality of certain classes of numbers. Amer. Math. Monthly. — 1954. — 61, No. 4. — P. 235-241. [19] Rucki P. The irrationality of infinite series of a special kind. Math. Slovaca. — 2016. — 66, No. 4. — P. 803–810. [20] Serbenyuk S. O. Real numbers representation by the Cantor series. International Conference on Algebra dedicated to 100th anniversary of S. M. Chernikov: Abstracts, Kyiv: Dragomanov National Pedagogical University, p. 136 (2012). Link: https://www.researchgate.net/publication/311415815, https://www.researchgate.net/publication/301849984 [21] Serbenyuk S. O. Cantor series expansion of real numbers: expansion of rational numbers. Naukovyi Chasopys NPU im. M. P. Dragomanova. Ser. 1. Phizyko-matematychni Nauky [Trans. Natl. Pedagog. Mykhailo Dragomanov Univ. Ser. 1. Phys. Math.] — 2013. — 14. — P. 253-267. (Ukrainian) Link: https://www.researchgate.net/publication/283909906 [22] Serbenyuk S. Cantor series and rational numbers, available at https://arxiv.org/pdf/1702.00471.pdf [23] Serbenyuk S. Representation of real numbers by the alternating Cantor series, http://arxiv.org/pdf/1602.00743v1.pdf. [24] Sondow J. A geometric proof that e is irrational and a new measure of its irrationality. Amer. Math. Monthly. — 2006. — 113. — P. 637–641. [25] Tijdeman Robert, Pingzhi Yuan. On the rationality of Cantor and Ahmes series. Indag. Math. (N.S.). — 2002. — 13, 3. — P. 407-418. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska St., Kyiv, 01004, Ukraine E-mail address: [email protected]