S. Yu. Favorov, N. P. Girya ON A PROPERTY OF PAIRS OF ALMOST PERIODIC ZERO SETS

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1 Математичнi Студiї. Т.43, 1 Matematychni Studii. V.43, No.1 УДК S. Yu. Favorov, N. P. Girya ON A PROPERTY OF PAIRS OF ALMOST PERIODIC ZERO SETS S. Yu. Favorov, N. P. Girya. On a property of pairs of almost periodic zero sets, Mat. Stud. 43 (2015), Whenever all differences of zeros of two holomorphic almost periodic functions in a strip form a discrete set, then both functions are infinite products of periodic functions with commensurable periods. In particular, the result is valid for some classes of Dirichlet series. С. Ю. Фаворов, Н. П. Гиря. О свойстве пар почти периодических нулевых множеств. // Мат. Студiї Т.43, 1. C Доказано, что если множество разностей нулей двух голоморфных почти периодических функций в полосе дискретно, то обе функции являются бесконечным произведением периодических функций с соизмеримыми периодами. В частности, результат справедлив для некоторых классов рядов Дирихле. It was proved in [1] that each quasipolynomial Q(z) = N a n e λnz, λ n R, a n C, (1) n=1 with a discrete set of differences of its zeros is periodic up to a multiplier without zeros, consequently it has the form Q(z) = Ce βz N k=1 cosh(ωz + b k ), β, ω R, C, b k C. (2) The result is also valid for infinite sums S(z) = a n e λnz, λ n R, a n C, (3) n=1 under conditions a n <, λ 1 = sup λ n <, n n=1 λ 2 = inf n λ n >, a 1 a 2 0. (4) Note that zeros of functions (1), (3) are located in a vertical strip of a finite width Mathematics Subject Classification: 30B50, 42A75. Keywords: quasipolynomial; periodic function; zero set; Dirichlet series; almost periodic holomorphic function. doi: /ms c S. Yu. Favorov, N. P. Girya, 2015

2 44 S. Yu. FAVOROV, N. P. GIRYA In 1949 M. G. Krein and B. Ja. Levin (see [5], also [6], Ch. 6, p. 2 and Appendix 6) introduced and studied the class of entire almost periodic functions of exponential growth with zeros in a horizontal strip of a finite width. In particular, if the sum S(z) in (3) satisfies condition (4), then S(iz) belongs to. In fact, representation (2) was obtained in [1] for functions from with a discrete sets of differences of zeros (of course, one should replace e βz by e iβz and cosh(ωz + b k ) by cos(ωz + b k )). The similar phenomenon takes place for a pair of functions when the set of differences of their zeros is discrete ([3]). For example, let Q 1, Q 2 be entire functions of form (3) under conditions (4). If the set {z w, Q 1 (z) = 0, Q 2 (w) = 0} is discrete, then the both functions have form (2) with the same ω and possibly different C, β, N, b k. In particular, in the case Q 2 ( z) = Q 1 (z) = Q(z) we get representation (2) for any function (3) with the discrete set {z + z, Q(z) = Q(z ) = 0}. The proof of the above results are based on the property of zeros {z j } of functions f ([6], Appendix 6, p. 2) to be almost periodic in the sense of the following definition. Definition 1. A zero set {z j } is almost periodic if for any ε > 0 there is a relatively dense set E ε R such that for each τ E ε there exists a bijection σ : N N with the property sup z j + iτ z σ(j) < ε. (5) j Recall that a set E R is relatively dense if there exists L < such that E [a, a+l] for any a R. In a sense, the above definition is applied to zero sets in a closed vertical strip. In the present paper we investigate holomorphic almost periodic functions and almost periodic sets in an open strip, in particular, in the complex plane and in half-planes. Definition 2. A continuous function f(z) in the strip S = {z : a < Re z < b}, a < b is almost periodic, if for any substrip S 0, S 0 S, and any ε > 0 there is a relatively dense set E ε,s0 R such that for any τ E ε,s0 sup z S 0 f(z + iτ) f(z) < ε. (6) A typical example of an entire almost periodic function is sum (3) under conditions n=1 a n <, sup λ n <, n inf n λ n >. Note that a holomorphic function f(z) in the strip S = {z : a < Re z < b}, a < b is almost periodic if and only if there exists a sequence of quasipolynomials Q n of form (1) such that for any substrip S 0, S 0 S, sup f(z) Q n (z) 0, n, z S 0 (see [4], item 8). In order to take into account multiplicities of zeros, we use the term divisor instead of zero set. Namely, a divisor Z in a domain D is the mapping Z : D N {0} such that Z = suppz is a set without limit points in D. In other words, a divisor in D is the sequence

3 ON A PROPERTY OF PAIRS OF ALMOST PERIODIC ZERO SETS 45 of points {z j } D without limit points in D such that every point of D appears at most finite times in the sequence. If Z(z) 1 for all z D, we identify Z and Z. The divisor of zeros of a holomorphic function f in D is the map Z f such that Z f (a) equals the multiplicity of zero of the function f at the point a. The following definition appeared at first in [8]. Definition 3 ([2], [8]). A divisor Z = {z j } in a strip S = {z : a < Re z < b}, a < b, is called almost periodic if for any ε > 0 and any substrip S 0, S 0 S, there is a relatively dense set E ε,s0 = {τ R: z j S 0 z σ(j) S 0 = z j + iτ z σ(j) < ε}, (7) where σ = σ τ is a suitable bijection N N. Remark. If we take σ (j) instead of j in (7), we get z j S 0 z σ (j) S 0 = z j iτ z σ (j) < ε. (8) Theorem 1 ([2]). a) If f is a holomorphic function in a strip S with the almost periodic modulus, then the divisor of zeros of f is almost periodic. b) For any almost periodic divisor Z in a strip S there is the holomorphic function f in S with the almost periodic modulus such that Z = Z f c) For any almost periodic divisor Z in a strip S such that Z S = for some open substrip S S there is the holomorphic almost periodic function f in S such that Z = Z f. There exists an almost periodic divisor in the plane with a discrete set of differences of their points that is not periodic. Example 1. Let Z = {z n,k = i2 nk + 2 k, n Z, k N} be a discrete set in C. It is easy to see that the set is almost periodic, differences of its points form a discrete set, but Z is a countable union of periodic sets with different commensurable periods. In our article we prove the following theorem: Theorem 2. Let Z, W be almost periodic divisors in the strip S = {z : a < Re z < b}, a < b. If a) Z S = for some substrip S S, b) for any substrip S 0 of a finite width, S 0 S, the set {z w, z Z S 0, w W S 0 } is discrete, then Z, W are at most countable sums of periodic divisors with commensurable periods. Show that condition a) is essential. Example 2. Let S = {z : Re z < 1}, Z = {z m,n = (m + in)e iα S, m, n Z}, where α R such that cot α is an irrational number. Clearly, the set of differences of elements of Z is discrete. Let us prove that Z is an almost periodic divisor without any periods. By Kronecker Lemma (see, for example, [7], Ch.2, 2), for any ε > 0 the inequalities exp(2πit cot α) 1 < ε, exp(2πit) 1 < ε,

4 46 S. Yu. FAVOROV, N. P. GIRYA has a relatively dense set of common solutions. Therefore, the inequality exp(2πim cot α) 1 < ε(1 + cot α ) has a relatively dense set of integer solutions. The latter means that for any δ there exist pairs of integers (m, n) Z 2 such that and the set of m with this property is relatively dense. Put m cos α n sin α < δ, (9) E = {m sin α + n cos α: m cos α n sin α < δ}. Let τ = m sin α + ñ cos α E. For any z m,n Z and m = m + m, n = n + ñ we have z m,n + iτ z m,n = (m + in)eiα + i Im[( m + iñ)e iα ] (m + in )e iα = Re[( m + iñ)e iα ]. By (9), z m,n + iτ z m,n < δ. Since τ (cos 2 α/ sin α + sin α) m = ñ cos α (cos 2 α/ sin α) m < δ cot α, we see that the set E is relatively dense. So, Z is an almost periodic divisor. Since Z sites in the vertical strip of width 2, we see that any period of Z must have the form it, T R. Hence, for some (n, m), (n, m ) Z 2, (n, m) (n, m ), (n + im)e iα (n + im )e iα = it, (n n ) + i(m m ) = it (cos α i sin α). This equality contradicts our choice of α. Hence any part of Z has no periods. Our proof of Theorem 2 uses the following lemmas. Lemma 1. Let Z = {z n } be an almost periodic divisor in a strip S, S 0 be a substrip, S 0 S, ε > 0, τ 1, τ 2 E ε,s0, where E ε,s0 is from Definition 3. Then there is a bijection σ : N N such that if z j S 0, dist(z j, S 0 ) > ε, then z j + i(τ 1 τ 2 ) z σ(j) < 2ε, z j i(τ 1 τ 2 ) z σ (j) < 2ε. (10) It can be proved easily that the similar assertion is valid for τ 1 + τ 2 as well. Proof. If z j satisfies conditions of the lemma, then for some bijection σ 1 : N N we have z j + iτ 1 z σ1 (j) < ε, therefore, Re(z σ1 (j) z j ) < ε and z σ1 (j) S 0. By (8), we also get the inequality z σ 2 σ 1(j) iτ 2 z σ1 (j) < ε for some bijection σ 2 : N N. Hence we obtain the first part in (10) with σ = σ2 σ 1. If we change places of τ 1 and τ 2, we obtain the second inequality in (10). Lemma 2. Let Z = {z n }, W = {w m } be almost periodic divisors in a strip S. Then for any ε > 0 and any substrip S 0 S such that dist(s 0, S) > ε, there is a relatively dense set E R with the property: for any τ E there are bijections σ Z, σ W from N to N such that for all z j Z S 0, w r W S 0 z j + iτ z σz (j) < ε, w r + iτ w σw (r) < ε, (11) z j iτ z σ (j) < ε, w r iτ w Z σ W (r) < ε. (12)

5 ON A PROPERTY OF PAIRS OF ALMOST PERIODIC ZERO SETS 47 Proof. Put S = {z S : dist(z, S 0 ) < ε/2}. By (7) and (8), there is a number L such that any interval of length L contains τ Z and τ W with the properties for all z j Z S, and z j + iτ Z z σz (j) < ε/4, z j iτ Z z σ Z (j) < ε/4 (13) w r + iτ W w σw (r) < ε/4, w r iτ W w σ W (r) < ε/4 (14) for all w r W S. Here σ Z, σ W are some bijections N N. We may suppose that N = 2L/ε is an integer. Hence for any k Z there are integers n(k), m(k), 0 n(k), m(k) N, such that kl + n(k)ε/2 τ Z < ε/4, kl + m(k)ε/2 τ W < ε/4. The differences n(k) m(k) take at most 2N + 1 values. Choose k 1,..., k r, r 2N + 1, such that for any k Z there is k s with the property n(k) m(k) = n(k s ) m(k s ). It follows easily that any interval of length L(max s ( k s + 2) contains a point of the form τ = Lk + n(k)ε/2 Lk s n(k s )ε/2 = Lk + m(k)ε/2 Lk s m(k s )ε/2. If we replace τ Z by Lk + n(k)ε/2 or Lk s + n(k s )ε/2 in (13), we obtain that this inequality satisfies with ε/2 instead of ε/4. The same is true if we replace τ W by Lk + m(k)ε/2 or Lk s + m(k s )ε/2 in (14). Applying Lemma 1 with S 0 = S and ε/2 instead of ε, we see that τ satisfies (11) and (12). Proof of Theorem 2. Set a sequence of vertical open substrips S k, S k S k+1, such that Z S 1, W S 1, S 1 S, S k = S. Take any points z Z S 1, w W S 1. It follows from (11) with ε < min{dist(z, S 1 ), dist(w, S 1 )} that there is R < such that any horizontal strip of the width R contains at least one point Z S 1 and at least one point W S 1. Let δ be less than the width of the strip S. By condition b) of the theorem, for each k the set {z w, z Z S k, w W S k } is discrete. Hence there exists γ k < min{1/2, δ, dist(s k, S k+1 )} such that whenever z n, z n Z S k+1, w m, w m Z S k+1, z n w m z n w m, Im(z n w m ) < 2R + 3, Im(z n w m ) < 2R + 3, we get γ k < (z n w m ) (z n w m ). In particular, if we put w m = w m, then we get γ k < z n z n for any z n, z n Z S k+1, z n z n. Fix z n Z S k, and let a number τ > 1 satisfies (11) for Z and W with ε = γ k /2. Then there is a unique z n Z S k+1 such that z n + iτ z n < γ k /2. Indeed, otherwise we obtain z n z n z n + iτ z n + z n + iτ z n < γ k. Set T k = (z n z n )/i. First suppose that T k is real, hence, z n Z S k. k

6 48 S. Yu. FAVOROV, N. P. GIRYA Let w m W S k be such that Im (w m z n ) < 2R + 2. By (11), there is a point w m W S k+1 such that w m + iτ w m < γ k /2. Therefore, Since (z n w m ) (z n w m ) w m + iτ w m + z n z n iτ < γ k. Im(z n w m ) Im(z n w m ) + z n z n + iτ + w m w m + iτ < 2R + 3, we get z n w m = z n w m due to the choice of γ k. Therefore, w m = w m + it k, w m S k. The latter equality takes place for all points W {w S k : Im z n 2R < Im w < Im z n + 2R}, in particular, for some w l such that Im z n + R < Im w l < Im z n + 2R. Namely, there is w l W S k such that w l = w l + it k. Let ζ Z S k be any point from the set {z S k : Im z n Im z < Im z n + 3R} {z S k : Im w l 2R < Im z < Im w l + 2R}. (15) By (11), there is a point ζ Z S k+1 such that ζ + iτ ζ < γ k /2. Therefore, Since Im ζ Im w l < 2R and (ζ w l ) (ζ w l ) ζ + iτ ζ + it k iτ < γ k. Im(ζ w l ) Im(ζ w l ) + ζ + iτ ζ + it k iτ < 2R + 3, we get ζ w l = ζ w l due to the choice of γ k. Therefore, ζ = ζ + it k. In particular, there is a point z s Z {z S k : Im z n + 2R Im z < Im z n + 3R} such that z s + it k Z. Continuing the line of reasoning, we obtain that for all z Z such that Im z Im z n we get z + it k Z and for all w W such that Im w Im z n we get w + it k W. If we take w l W S k such that Im z n 2R < Im w l < Im z n R, we also can find w l W such that w l = w l + it k. Next, we prove that for any point ζ Z {z S k : Im z n 3R Im z < Im z n } there is ζ Z S k such that ζ = ζ + it k. Arguing as above, we show that for all z Z S k such that Im z Im z n we get z + it k Z and for all w W S k such that Im w Im z n we get w + it k W. Next, by (12), take for any z Z S k a point z Z S k+1 such that z +iτ z < γ k /2. Then z + it k Z S k and (z + it k ) z z + iτ z + iτ it k < γ k. Therefore, z + it k = z and z it k Z for all z Z S k. By the same arguments, w it k W for all w W S k. Now suppose that Im T k 0. Clearly, either dist{z n, S } > dist{z n + it k, S }, or dist{z n, S } > dist{z n it k, S }. In the first case the points z n, w l, w l ζ, ζ... belong to S k. Hence, we get z n + imt k S for some integer M > 0, that is impossible. In the second case the same arguments show that z n + imt k S for some integer M < 0. We get a contradiction in the both cases. Therefore, T k is real and for any M Z we get ( Z S k )+imt k = Z S k, ( W S k )+imt k = W S k. Thus the restrictions Z Sk, W Sk are periodic divisors with period it k. The same arguments work for every k {1, 2,... }. Let itk 0 be the minimal common period of Z S k and W Sk. Clearly, T k /Tk 0 N. Besides, since Z S k Z S m for m > k, we have T m /Tk 0 N as well. Finally, let Z 1 = Z S1, Z k = Z Sk \S k, W 1 = W S1, W k = W Sk \S k. Then we obtain Z = Z 1 + Z 2 + Z , W = W 1 + W 2 + W

7 ON A PROPERTY OF PAIRS OF ALMOST PERIODIC ZERO SETS 49 Theorem 2 implies the corresponding result for almost periodic holomorphic functions. Theorem 3. Let f, g be almost periodic functions in a strip S = {z : a < Re z < b}, a < b. If a) either f, or g has no zeros in an open substrip S S, b) for any substrip S 0, S 0 S, the set {z w, z, w S 0, f(z) = g(w) = 0} is discrete, then f(z) = f 0 (z) f k (z), k=1 g(z) = g 0 (z) g k (z), (16) where f 0, g 0 are holomorphic almost periodic functions in S without zeros, f k, g k, k {1, 2,... }, are periodic holomorphic in S with commensurable periods it k. Proof. By Theorem 1a), the divisors Z, W of zeros of f, g, respectively, are almost periodic and satisfy other conditions of the previous theorem. Let S k, Z k, W k, be the same as in the proof of Theorem 2. Suppose that k=1 S k = {z : η k < Re z < η k}, η k < η k, η k > η k, k. For any k there is only a finite number of points a k 1,..., a k m k Z k {z : 0 Im z < T k }. Therefore, Z k = {a k 1,..., a k m k } + it k Z. Take ε j k > 0, 1 j m k, 3 k <, such that <. (17) Set j,k ε j k h j (w) = 1 w exp( 2πa k j /T k ), k {1, 2,...}, j {1,..., m k }. Clearly, h j (exp(2πz/t k )) has the divisor a k j + it k Z. Now let k > 2. Since αj k S k \ S k, we see that either η k Re ak j, or η k Re a k j. In the first case, log h j (w) is holomorphic on the disc w < exp(2π Re a k j /T k ), and there is a polynomial Pj k such that log h j (w) P k j (w) < ε j k for w exp(2πη k 2/T k ). In the second one, log h j (w) is holomorphic on the set w > exp(2π Re a k j /T k ), and there is a polynomial Pj k such that log h j (w) P k j (w ) < ε j k for w exp(2πη k 2 /T k ). Put Q j (z) = P k j (exp(2πz/t k )) in the first case, and Q j (z) = P k j (exp( 2πz/T k )) in the second one. We obtain Put h j (e 2πz/T k )e Q j(z) < e εk j for z S k 2. (18) m k m k f k (z) = h j (e 2πz/T k ), k = 1, 2, f k (z) = h j (e 2πz/T k )e Qj(z), k > 2. j=1 Clearly, Z fk = Z k. By (17) and (18), the first product in (16) converges in every substrip S k. The function f(z)/ k=1 f k(z) is holomorphic in S, and, by [8], almost periodic in S. In the same way, we obtain the representation of g. Remark. It follows easily that for entire almost periodic functions f, g with zeros in a strip S of a finite width one can take S 1 = S. Therefore, f, g are periodic functions with the same period up to almost periodic multiplies without zeros. Hence some results of [1], [3] follow from Theorem 3. j=1

8 50 S. Yu. FAVOROV, N. P. GIRYA REFERENCES 1. S.Ju. Favorov, N.P. Girya, On a criterion of periodicity of polynomials, Ufa s Mathematical Journal, 4 (2012), 1, S.Yu. Favorov, A.Yu. Rashkovskii, A.I. Ronkin, Almost periodic divisors in a strip, J. d Analyse Math., 74 (1998), N.P. Girya, Periodicity of Dirichlet series, Mat. Stud., 42 (2014), 1, B. Jessen, H. Tornehave, Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), M.G. Krein, B.Ja. Levin, On entire almost periodic functions of an exponential type, DAN SSSR, 64 (1949), 2, (in Russian) 6. B.Ja. Levin, Distributions of zeros of entire functions, V.5, Transl. of Math. Monograph, AMS Providence, R1, B.M. Levitan, Almost periodic functions, Gostehizdat, Moscow, (in Russian) 8. H. Tornehave, Systems of zeros of holomorphic almost periodic functions, Kobenhavns Universitet Matematisk Institut, Preprint, 30, Karazin Kharkiv National University sfavorov@gmail.com n girya@mail.ru Received Revised