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21Probl. Anal. Issues Anal. Vol. 8 (26), No3, 2019, pp. 63-72
63
DOI: 10.15393/j3.art.2019.6470 UDC 517.52, 517.53/.55
A. F. Kuzhaev
ON THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE MEASURABILITY OF A POSITIVE SEQUENCE
Abstract. The work is devoted to finding out the necessary and sufficient conditions for the measurability of a sequence of positive numbers. The concept of logarithmic measurability of a sequence is also introduced. It is assumed that the considered sequences form a sequence of zeros of some entire function of exponential type. Therefore, clarification of this question can be useful in solving the problem of completeness of the system of exponents or exponential monomials in some convex domain. Such characteristics of the sequence as lower and upper densities, minimum and maximum densities, lower and upper logarithmic block densities play an important role.
Key words: upper density, maximal density, logarithmic block density, zeros of entire function
2010 Mathematical Subject Classification: 30D10, 40B05
Let A = (An be an unbounded nondecreasing (numbers An can be repeated) sequence of positive numbers.
The symbol B(z, t) denotes a closed circle with the center at the point z and radius t. The upper and lower densities of the sequence A are, respectively, the following quantities:
-/an v— n(t, A) . n(t, A)
n(A) = lim v ', n(A) = lim v ', t t
where n(t, A) is the counting function of the sequence A, that is, the number of its elements counting the multiplicities located in the circle B(z, t):
n(t, A) = £ 1.
© Petrozavodsk State University, 2019
If n(A) = n(A), the sequence A is called measurable and the quantity
/AN T n(t, A) n(A) = lim v '
t^ + x t
exists and is called the density of the sequence A.
We recall that the maximal and minimal density of the sequence A are, respectively, the quantities
-/an i- T-— n(t, A) - n(t(1 - S), A)
n0(A) = lim lim v ' '-^-, S e (0; 1).
0V ; t^+x St v ;
/AN T T n(t,A) - n(t(1 - S),A)
n0(A) = lim lim v ' '-^-, S e (0; 1).
t^+x St
According to the Lemma in Section E3, Chapter VI in [3], the limit as S ^ 0+ always exists and the maximal density is well-defined.
If the sequence A is a sequence of positive numbers, then the logarithmic block density, introduced in [6], is:
L(A) = inf Um A(at) - A(t), A(t) = V ^.
«>i t^+x ln a An
K^t n
According to Lemma 3.2 of [5], the quantity L(A) can be calculated as follows:
L(A) = lim Um A(at) - A(t),
a^+x t^+x ln a
that is, the limit as a ^ ^ exists. Making the change of variables in the latter identity, we can write
7/AN v A(t) - A(t(1 - S))
L(A)= lim lim v ' \ v - n, ¿^1-0 t^+x - ln(1 - S)
where S e (0; 1). In what follows, we shall employ exactly this identity while working with the logarithmic density.
For an arbitrary S e (0; 1), consider the following quantities
A(t) - A(t(1 - S)) L(A , St)= - ln(1 - S) ,
L(A ,S) = um L(A ,S,t) , L(A ,S) = lim L(A ,S,t) ,
t^+x t^+x
L(A,0) = lim L(A,0,t),
t—
and, likewise,
no(A,il() = "<'■A) - "f1 - *>■A),
ot
no(A,0) = lim no(A,0,t), no(A,0)= lim no(A,0,t), t—t—
no(A,0)= lim n0(A, 0, t). t—
It is clear that
n0(A) = lim no(A,0), n0(A) = lim no(A,0),
5—+0 5—+o
L(A)= lim L(A,0).
5—1-0
Later in this paper, the quantity L(A) will be called the upper logarithmic block density of the sequence A in order to distinguish it from the lower logarithmic block density, which we define as
L(A)= sup L(A,0).
5e(0;1)
Lemma 1. The function L(A,0) has a limit at 0 ^ 1 — 0, and the following equality holds:
L(A)= lim L(A, 0).
_V y 5-S-1-0
Proof. We reason as in the proof of Lemma 3.2 of [5]. Consider, for u > 0, the function
p(u) = lim (A(eur) - A(r)). t—^ +
Since eu2r ^ ^ at r ^ ^ for any fixed u2 > 0, it follows from the properties of the lower limit that
p(ui + U2) = lim (A(eui+u2r) - A(r)) = t—
= lim (A(eui +u2r) - A(eu2r) + A(eu2r) - A(r)) ^ t—
^ lim (A(eui+u2r) - A(eu2r)) + lim (A(eu2r) - A(eur)) = p(ui) + <^2). t—t—
Let v > 1 be fixed, u be large enough; then there is such a positive integer N that u = Nv + w, where 0 ^ w < r. In this case
<^(u) ^ N^(v) + <^(w) <^(v) 1 ^ ^(w)
u Nv + w v 1 + w/Nv Nv + w
Since N ^ at u ^ at any v > 1 we have
lim ^ul ^ .
t^+x u v
It follows from this inequality that
lim - ^ lim -, km - ^ sup-.
u u u u>1 u
Since inequalities with the opposite signs are obviously true,
^(u)
lim -= sup-.
M^+x u u> 1 u
Making the change of variables u = — ln(1 — S), r = t(1 — S), S G (0; 1), we obtain
lim Hm A(t) — A(t(1 — S)) = lim L(A, S) = sup L(A,S) = L(A). <M-i-o t^+x — ln(1 — S) <s^1-o ¿e(o;1)
The lemma is proved. □
Exploring relationships between different densities is of the most interest in the case when the sequence A is a sequence of zeros of some entire function of exponential type. According to Lindelof's theorem, the sequence A must have a finite upper density. Therefore, all the reasonings given below can be carried out assuming that n(A) < to.
Let S G (0; 1); then, according to Lemma 1 from [4] and Lemma 2.1 from [1], the series of inequalities is valid:
n0(A) ^ n(A, S) ^ n(A) ^ L(A) ^ n(A) ^ no (A, S) ^ no(A) (1)
If the sequence A is measurable, then all inequalities in this chain pass into equalities.
It is easy to see that for any sequence A the inequality holds:
L(A) ^ L(A).
Lemma 2. The following inequality holds:
n(A) ^ L(A).
Proof. If n(A) = 0, then the statement is trivial. Consider the case n(A) > 0. Since n(A) < to, then, by (1), we have n(A) < to. Hence, for any sufficiently small e > 0 there is r£ such that for r ^ r£ the inequalities hold:
n(r, A) ^ (n(A) — e)r, n(r, A) ^ (n(A) + e)r. Let S G (0; 1) and t(1 — S) ^ re. We have:
t
^ 1 _ f dn(r, A) _ n(t, A) n(t(1 — S), A) |
Xn J r t t(1 - S)
i(1-5)<A„<i n t(1-5) v '
t t f n(r, A)dr n(t, A) n(t(1 - S), A) . ... . f dr
+ J "V" - Vs) ' +<n(A) - e) / " >
t(1-5) t(1-5)
t
n(t(1 - S), A) . ,4. . f dr
> - ^k-S) + <"<A) - e) J V>
t(1-<S)
> -(-(A) + e) + (-(A) - e)ln' 1
1 - S
Hence, based on Lemma 1 and letting S ^ 1 - 0, we obtain the desired result. The lemma is proved. □
Lemma 3. The equality holds:
-0(A) = lim L(A,S).
Proof. Let S G (0; 1). Then
nft,A) - - S)-A) ^ 1 y ± =
v 7 St S ^ Xn
t(1S)<Xn^t n
= - ln(1 - S) £<A,S,t).
S
From here we obtain
Likewise,
n0(A,0) ^ ln(1 0) L(A,0). 0
L<A^> = -n^ £ mn <
v 1 t(1-5)<An<t n
^ 1 n(t, A) - n(t(1 - 0), A) ^ 0 _ (A o t)
^ -ln(1 - 0) (T-ty ^ (0 - 1)ln(1 - 0)_o(A,0,t).
Therefore,
£(A'0) < (0 - 1)ln(1 - 0)'_o<A-0) (3)
Moving in (2) and (3) to the limit at 0 ^ 0+, we obtain the lemma statement. □
We will say that the positive sequence A is logarithmically measurable if L(A) = L(A).
The following statement is true, which is the main result of this work.
Theorem 1. Let A = (Anj^=1 be a unbounded nondecreasing sequence of positive numbers. Then the following statements are equivalent:
1) the sequence A is measurable;
2) the quantity L(A, 0) exists and does not depend on 0 G (0; 1);
3) the quantity n0(A,0) exists and does not depend on 0 G (0; 1).
Proof. The implication 1) 2) follows from Lemma 2 and the chain of inequalities (1).
If the quantity L(A,0) exists and does not depend on 0 G (0; 1), then it follows from Lemma 3 and [4, Theorem 1] that n0(A) = n0(A). This gives us an implication 2) 3). Finally, if the quantity n0(A,0) exists and does not depend on 0, then we obtain that the maximal density is equal to the minimal density, and then from the chain of inequalities (1) the measurability of the sequence A follows. This gives an implication 3) 1). The proof is complete. □
Remark 1. We give an example of a sequence that shows that logarithmic measurability not entails measurability. Consider a sequence of natural numbers An G N satisfying the condition
An+1
An
—> oo, n —oo. (4)
Suppose that the multiplicity of each point An is = An. Let 5 G (0; 1) be fixed. Then, by virtue of (4), there is such a number N0 that for any n ^ N0 the half-interval of the form (An(1 — 5), An] contains only the point An. It follows that at any t > 0 the interval of the form (t(1 — 5), t] contains no more than one point of sequence An. We have
A(t) — A(t(1 — 5)) — ln(1 — 5)
£
An ^
— ln(1 — 5) ^ An
^ ' t(1—¿)<An<t n
— ln(1 — 5)'
From here we get that L(A) = L(A) = 0. Now calculate the upper and lower densities. Since for any n ^ 1 and for a sufficiently small £ > 0, inequalities
n(An — £, A) < n(An, A) n(An + £, A) < n(An, A)
An £
An
An + £
An
are true; so,
n(A) = Hm n(An,A), „(a) = lim n(An — £,A) = lim n(An — £,A)
An
n—^^o s^n
An £
An
We have
-rw n(An, A) — Ai + A2 +-----+ A„
n(A) = lim -;-= lim
n—^ An
An
1 I r- ( A1 I A2 , , An-1
1 + lim — + — +-----+ ——
V An An An
It follows that n(A) ^ 1. On the other hand, by virtue of (4), for any number M > 1 there exists a number n0 such that for all n ^ n0 the inequality
AA
(5)
An+1 > M or 1 < 1/M, n ^ n0
An An
is satisfied. Since the dropping of a finite number of sequence points does not affect the value of its densities, it can be assumed, without loss of generality, that (5) is fulfilled for all n ^ 1. Then
Ai A2
0 ^ r + f +
An An
+
An— 1
A1 A2 A2 A3
n
An— 1
An
+A3
A«- 1
An
+
+ 1 < An
1
1
1 1 1
< Mn-1 + Mn—2 + "' + M
1 1--7T"—7 ) ^ 0, M ^ + TO.
/1 An_ J '
m -1 V Mn-1
From here we get n(A) ^ 1. Therefore, n(A) = 1. At the same time, n(A) = lim n(An - £,A) = lim f A" + A- + ... + ) =0.
n^-x An n^-x \ An An An /
Hence, n(A) < n(A). This means that the A sequence is not measurable, though L(A) = L(A).
Remark 2. A natural question arises: can this result be extended to complex sequences? It turned out that the answer is no. For such sequences the upper logarithmic block density is defined as
l(A) = inf nm /(aM'A)
«>i t^+x ln a
where
Z(ti,t2, A) = max{/"(ii ,¿2, A), /+(¿1, ¿2, A)},
1
An
(ti, ¿2, A) = ^ - Re A-,
tl<|A„Kt2,Re A„<0
1
A„
/+(¿i, ¿2, A)= J] Re —,
tl<|A„|^t2,Re An>0
(see [2], [7], [8]). This is explained by the fact that the chain of inequalities similar to the chain (1) and the complex version of Lemma 2 are necessary for the validity of the theorem statement in the complex case. However, Lemma 2 is no longer valid in the case of a complex sequence. For example, for the sequence A = {^v/n}X=i we obtain the following equations:
n(A) = lim =r^" = lim —^ = to, L(A) = 0.
n^x |An| n^x V n
This is a counterexample to Lemma 2 in the complex case. Note that in [9, Theorem 3, Theorem 6] there are more general statements where it is shown that the upper density of a positive sequence of points can be arbitrarily large, while the logarithmic block densities in all directions are
zero. From [9, Theorem 6] it can be seen that the upper density can be replaced by the lower density arbitrarily large.
Acknowledgment. This work was supported by a grant from the Russian science Foundation (project No. 18-11-00002).
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DOI: https://doi .org/10.4213/sm3885
Received June 13, 2019. In revised form,, August 21, 2019. Accepted August 23, 2019. Published online September 13, 2019.
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Bashkir State University
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