Trigonometric series with noninteger harmonics
aa r X i v : . [ m a t h . C A ] F e b TRIGONOMETRIC SERIES WITH NONINTEGER HARMONICS
MIKHAIL R. GABDULLINA
BSTRACT . Let t c k u be a nonincreasing sequence of positive numbers, and α ą be not aninteger. We find necessary and sufficient conditions for the uniform convergence of the series ř k c k sin k α x and ř k c k cos k α x on the real line and its bounded subsets.
1. I
NTRODUCTION
Let t c k u k “ be a nonincreasing sequence of positive numbers, and α ą . We investigatenecessary and sufficient conditions for the series(1.1) ÿ k “ c k sin k α x and(1.2) ÿ k “ c k cos k α x to converge uniformly on the real line R and its bounded subsets. Trivially, the condition ř k c k ă 8 is sufficient, but can one weaken it? This question for the sine series (1.1) wasconsidered in several papers. First, more than a century ago it was shown in the work [CJ] thatin the case α “ the series (1.1) converges uniformly on r , π s (equivalently, on the real line R ) if and only if c k k Ñ as k Ñ 8 (note that, under assumption of monotonicity, ř k c k ă 8 implies c k k Ñ ; see Lemma 2.1). Since then, many generalizations of this theorem have alsobeen obtained, where the same was shown for more general classes of sequences t c k u (that is,the requirement of its monotonicity was relaxed); see, for instance, the works [Nur], [Ste], [Lei],[Tik], and also [Og] for an overview of such results. In what follows, we consider nonincreasingsequences t c k u only.Further, in the recent paper [Kes] it was proved that in the case α “ { the condition c k k Ñ , k Ñ 8 , is necessary and sufficient for the uniform convergence of the series (1.1) on r , π s , and that in the case α “ it converges uniformly on r , π s (equivalently, on R ) if andonly if ř k c k ă 8 ; the case α “ { is also discussed in the wonderful book [Sel] (see problemIV.5.10). As for the case of general α , a nice breakthrough was made in the work [Og]. Themain results of that paper are the following: firstly, for any odd α P N , the series (1.1) convergesuniformly on R if and only if c k k Ñ , k Ñ 8 (the same is shown for the series ř k sin f p k q x ,where f is an odd polynomial with rational coefficients); secondly, for any α P p , q the samecondition is equivalent to the uniform convergence of the series (1.1) on any bounded subset of R ; and thirdly, for any even α P N its uniform convergence on R is equivalent to the condition ř k c k ă 8 . Date : February 12, 2021.
Our contribution to this topic is the following three theorems. Firstly, for any noninteger α ą and ă a ă b we show that for the uniform convergence on r a, b s of the series (1.1) or(1.2) one can require something weaker than c k k Ñ , k Ñ 8 . Theorem 1.1.
Let t c k u be a nonincreasing sequence. Thena) for any α ą , α R N , there exists c p α q P p , q such that the condition ř k c k k ´ c p α q ă 8 is sufficient for the uniform convergence of the series (1.1) and (1.2) on r a, b s for any ă a ă b .Moreover, one can take c p α q “ α for α P p , q .b) for any α P p , q the condition c k k ´ α Ñ , k Ñ 8 , is necessary for the convergence ofthe series (1.1) or (1.2) at any point x ‰ . Note the condition in the part (a) is clearly weaker than c k k “ o p q (which is in fact thecriterion for the uniform convergence on bounded subsets of R ). The part (b) together withLemma 2.1 means that for α P p , q the condition ř k c k k ´ α ă 8 cannot be essentially relaxed.We did not try to optimize the value c p α q for α ą , and just mention that it is possible to take c p α q of order α ´ for large α (see Corollary 2.5).Secondly, we generalize the mentioned result from [Og] for any noninteger α ą . Theorem 1.2.
Let α ą be not an integer, and let t c k u be a nonincreasing sequence of positivenumbers. Then the series (1.1) converges uniformly on any bounded subset of R if and only if c k k Ñ as k Ñ 8 . Combining this with the results of [Og], we obtain criteria for the uniform convergence of(1.1) on the bounded subsets of the real line for any α ą .Finally, for rational noninteger α ą we find the criterion for the uniform convergence onthe whole real line. Theorem 1.3.
Let α ą , α P Q z N , and let t c k u be a nonincreasing sequence of positivenumbers. Then the series (1.1) converges uniformly on R if and only if ř k c k ă 8 . Recall that the same for even α ą was shown in [Og].In Section 2 we provide some auxiliary results. Sections 3, 4, 5 are devoted to the proofs ofTheorems 1.1, 1.2, 1.3 respectively. Theorems 1.1 follows from estimates for the exponentialsums ř M ď n ă M e πin α x ; the proof of Theorem 1.2 relies on it as well, but also needs boundsfor the sums ř M ď n ă M sin n α x , where x is small with respect to M . The proof of Theorem1.3 works for any α ą such that the square-free numbers (or at least a large part of them)to the degree α are linearly independent over Q . It is natural to conjecture that it is so for anynoninteger α , but this seems to be known only for noninteger rational ones. We mention thatthe analogues of Theorems 1.2 and 1.3 for the series (1.2) are trivial, since the convergence ofthis series at the point x “ implies that ř k c k ă 8 . Notation.
We use Vinogradov’s ! notation: F ! G (or G " F ) means that there exists aconstant C ą such that | F | ď CG . In many cases, this constant is allowed to depend on someparameters; say, the notation F ! α,a,b G means that there is a number C “ C p α, a, b q such that | F | ď C p α, a, b q G . We write F — G if G ! F ! G . We denote by t u u the largest integer notexceeding u . A positive integer n is said to be square-free if it has the form n “ p . . . p s , where p j are distinct primes. RIGONOMETRIC SERIES WITH NONINTEGER HARMONICS 3
Acknowledgements.
The author would like to thank Sergei Konyagin for introducing him tothis topic. The author is a Young Russian Mathematics award winner and would like to thank itssponsors and jury. This work was performed at the Steklov International Mathematical Centreand supported by the Ministry of Science and Higher Education of the Russian Federation(agreement no. 075-15-2019-1614).2. A
UXILIARY RESULTS
In this section we provide several technical results which we will rely on later. We begin withthe following simple lemma.
Lemma 2.1.
Let t u k u and t c k u be nonincreasing sequences of positive numbers. Then ř k c k u k ă 8 implies c k ku k Ñ as k Ñ 8 .Proof.
By the assumption, for any ε ą there exists k such that for all k ą k we have ř k { ă n ď k c n u n ă ε . But ř k { ă n ď k c n u n " c k ku k , and the claim follows. (cid:3) We will need the following well-known estimate for exponential sums due to van der Corputand Vinogradov. We use the standard notation e p y q “ e πiy . Lemma 2.2 (van der Corput) . Let f : r u, v s Ñ R be a function with θ ď | f p y q| ď ´ θ and f p y q ‰ for y P r u, v s . Then ÿ u ď n ă v e p f p n qq ! θ ´ . Proof.
See [IK], Corollary 8.11. (cid:3)
Lemma 2.3 (van der Corput) . Let r ě , M ě , and f : r M, M s Ñ R be a function suchthat Λ ď f p r q p y q ď η Λ for some Λ ą and η ě . Then ÿ M ď n ă M e p f p n qq ! η ´ r Λ γ M ` Λ ´ γ M ´ ´ r , where γ “ p r ´ q ´ .Proof. See [IK], Theorem 8.20. (cid:3)
Lemma 2.4 (I.M. Vinogradov) . Let M ě and f : r M, M s Ñ R be a smooth function suchthat for all j ě we have β ´ j F ď y j j ! | f p j q p y q| ď β j F, where F ě M and β ě . Then ÿ M ď n ă M e p f p n qq ! βM exp ` ´ ´ p log M q p log F q ´ ˘ . Proof.
See [IK], Theorem 8.25. (cid:3)
Corollary 2.5.
Let α ą be not an integer. Define the function c p α q “ $’&’% α, if α P p , q ;2 ´ t α u ´ , if α P p , q ;max t ´ t α u ´ , ´ α ´ u , if α ą . MIKHAIL R. GABDULLIN
Let ă a ă b . Then for all x P r a , b s and k ě we have k ÿ n “ e p n α x q ! α,a ,b k ´ c p α q . Proof.
Let α ą , α R N , and ă a ă b . Fix x P r a , b s and define f p y q “ xy α . It is enoughto show that for any M ě (2.1) ÿ M ď n ă M e p f p n qq ! α,a ,b M ´ c p α q . We may suppose that M is large enough depending on α, a , b , since otherwise the result fol-lows from taking the implied constant large on these parameters. We consider three cases.a) Let α P p , q . Since f p y q ‰ and | f p y q| “ αxy α ´ , we have | f p y q| ď αb p M q α ´ ď { and | f p y q| ě θ : “ αa M α ´ . Then (2.1) with c p α q “ α follows from Lemma 2.2.b) Let α ą . We apply Lemma 2.3 with r “ t α u ` ě ; then f p r q p y q “ α p α ´ q . . . p α ´ t α u ´ qp α ´ t α u ´ q xy α ´ t α u ´ ą clearly f p r q p y q — α,a ,b M α ´ t α u ´ “ : Λ . We have M ´ ď Λ ď M ´ and then ÿ M ď n ă M e p f p n qq ! α,a ,b M ´ γ ` M ` γ ´ ´ r . This implies (2.1) with c p α q “ ´ t α u ´ , since γ ě ´ r and ¨ ´ r ´ γ “ ´ r ` ´ r ´ ˘ ě ´ r as well.c) Let α ą . Then the condition of Lemma 2.4 holds with F “ M α and β is large enoughdepending on α, a , b . Thus we get (2.1) with c p α q “ ´ α ´ , as desired.This concludes the proof. (cid:3) Lemma 2.6.
Let u ă v be integers and f : r u, v s Ñ R be a monotone function. Then ÿ u ď n ă v f p n q “ ż vu f p y q dy ` O p max t| f p u q| , | f p v q|uq . Proof.
Without loss of generality we may assume that f is nondecreasing. Then for any integer n P r u, v s we have f p n q ď ş n ` n f p y q dy ď f p n ` q and hence ÿ u ď n ď v ´ f p n q ď ż vu f p y q dy ď ÿ u ` ď n ď v f p n q . The claim follows. (cid:3)
Lemma 2.7.
Let α ą , m P N , and x ą . Then I m p x q : “ ÿ p πmx q { α ď n ă p π p m ` q x q { α sin n α x ! α x ´ { α m { α ´ ` . Proof.
Set m “ r ` πmx ˘ { α s , m “ t ´ π p m ` { q x ¯ { α u , m “ r ´ π p m ` { q x ¯ { α s , m “ t ´ π p m ` { q x ¯ { α u , m “ r ´ π p m ` { q x ¯ { α s , m “ t ´ π p m ` q x ¯ { α u . We may suppose that m ´ m (and, hence, RIGONOMETRIC SERIES WITH NONINTEGER HARMONICS 5 m ´ m , m ´ m , and m ´ m ) is large enough, since otherwise the claim is trivial. Since thefunction sin y α x is bounded and monotone on each of the intervals r m , m s , r m , m s , r m , m s ,we see from the previous lemma that I m “ ż p π p m ` q x q { α p πmx q { α sin y α x dy ` O p q “ αx { α ż π p m ` q πm t { α ´ sin t dt ` O p q . Note that for any a ą ż π p m ` q πm t a sin tdt “ ż πm ` π πm t a sin tdt ` ż πm ` π πm ` π t a sin tdt “ ż π pp t ` πm q a ´ p t ` πm ` π q a q sin t dt ! a m a ´ . Thus I m ! α x ´ { α m { α ´ ` O p q , as desired. (cid:3) We see that the bound from Lemma 2.7 is nontrivial when α P p , q , or when α ě and x is small with respect to m . We will face the latter case in the proof of Theorem 1.2.We will also need another estimate for exponential sums. Lemma 2.8 (H.Weyl) . Let M ě , r ě and let a function f : r M, M s Ñ R be such that FB ď y r r ! | f p r q p y q| ď F for all y P r M, M s , where B ě and F ą . Then for any ď M ď M we have ÿ M ď n ď M ` M e p f p n qq ! B ´ r ` F M ´ r ` F ´ ˘ ´ r r ´ M log M. Proof.
See [IK], Theorem 8.4. (cid:3)
Lemma 2.9.
Let α ą be not an integer, x P p , q , and M P N be such that M ě x ´ {p α ´ δ q for some δ P p , α q . Then there exists absolute constant d p α, δ q P p , q such that for any ď M ď M ÿ M ď n ď M ` M e p n α x q ! α M ´ d p α,δ q . Proof.
We apply Lemma 2.8 with f p y q “ xy α , r “ t α u ` (say), F “ C xM α , and B “ C for appropriate constants C “ C p α q and C “ C p α q . Since x ´ ď M α ´ δ we then obtain ÿ M ď n ď M ` M e p n α x q ! α ` M ´ ` x ´ M ´ α ˘ ´ r r ´ M log M ! M ´ d p α,δ q for some d p α, δ q P p , q , as desired. (cid:3) MIKHAIL R. GABDULLIN
3. P
ROOF OF T HEOREM
Proof of the first statement of Theorem 1.1.
Fix α ą , α R N , and let c p α q be definedas in Corollary 2.5. Let t c k u k be a nonincreasing sequence with ř k c k k ´ c p α q ă 8 . Fix arbitrarypositive numbers a ă b ; we will prove that the series (1.1) and (1.2) converges uniformly on r a, b s . To do this, it is enough to show that for any ε ą there exists l “ l p ε q P N such that forall L ą l we have(3.1) sup x Pr a ,b s ˇˇˇˇˇ L ÿ k “ l c k e p k α x q ˇˇˇˇˇ ă ε, where a “ a {p π q , b “ b {p π q . Let b l “ sup k ě l c k k ´ c p α q and V k p x q “ ř kn “ e p n α x q . Thenusing consequently Abel’s summation, Corollary 2.5 and again Abel’s summation, for all x Pr a , b s we have (here the implied constants in ! are allowed to depend on α , a , and b ) ˇˇˇˇˇ L ÿ k “ l c k e p k α x q ˇˇˇˇˇ “ ˇˇˇˇˇ L ´ ÿ k “ l p c k ´ c k ` q V k p x q ´ c l V l ´ p x q ` c L V L p x q ˇˇˇˇˇ ! b l ` L ´ ÿ k “ l p c k ´ c k ` q| V k p x q|! b l ` L ´ ÿ k “ l p c k ´ c k ` q k ´ c p α q ! b l ` ÿ k ě l c k k ´ c p α q . Now (3.1) follows from Lemma 2.1 by taking l sufficiently large depending on α , a , b , and ε .This concludes the proof.3.2. Proof of the second statement of Theorem 1.1.
We first prove the claim for the series(1.1). Recall that a sequence t a n u is said to be uniformly distributed modulo if for any fixedinterval r c, d s Ď r , s we have t ď n ď N : a n p mod 1 q P p c, d qu “ p d ´ c ` o p qq N, N
Ñ 8 . Let α P p , q . Fix any x ‰ ; we may suppose that x ą . It is well-known (see, for example,[Mur], Exercise 11.6.3) that the sequence σn α is uniformly distributed modulo for any σ ‰ ;we take σ “ x {p π q . Now let m be large enough depending on α and x ; it follows that thereexists n “ n p m q P r m, m s such that n α x π P ˆ , ˙ p mod 1 q . Since for any r ą we have p n ` r q α ´ n α “ n α pp ` r { n q α ´ q “ n α p αr { n ` O p r { n qq “ αrn ´ α ` O ˆ r n ´ α ˙ , we see that p n ` r q α x π P ˆ , ˙ p mod 1 q for all r “ , , . . . , r “ t n ´ α π αx u (say). It means that for these r we have(3.2) p n ` r q α x P ˆ π , π ˙ p mod 2 π q . RIGONOMETRIC SERIES WITH NONINTEGER HARMONICS 7
On the other hand, if the series ř k c k sin k α x converges, then for any ε ą there exists m suchthat for all m ą m ą m the bound ˇˇˇˇˇ m ÿ k “ m c k sin k α x ˇˇˇˇˇ ă ε holds. Now we take any m ą m (again large enough depending on x and α ) and set m “ n “ n p m q , m “ n ` r ; then by (3.2) we get ε " n ` r ÿ k “ n c k " c m r " c m m ´ α , since m " α,x . It follows that c m m ´ α ! ε whenever m is large enough, and the claim follows.The proof for the case of the series (1.2) is completely similar; we just need to consider theinterval p´ { , q (say) instead of p { , { q , and then we get (3.2) with the interval p π , π q replaced by p´ π , π q . This concludes the proof of Theorem 1.1.4. P ROOF OF T HEOREM α ą be arbitrary, and suppose that theseries (1.1) converges uniformly on p , a q for some a ą . Then for any ε ą there exists l “ l p ε q such that for all L ą l sup x Pp ,a q ˇˇˇˇˇ L ÿ k “ l c k sin k α x ˇˇˇˇˇ ă ε. We can assume that l is large enough depending on α . Take L “ t { α l u and x “ πl ´ α { Pp , a q ; then L ď l and c l l ! ε ř Lk “ l c k ! ε , and hence c k k Ñ , k Ñ 8 , as desired.Now we prove that the condition is sufficient. Fix noninteger α ą , and let c k k “ o p q , k Ñ 8 . We need to show that the series (1.1) converges uniformly on any bounded subset of R .In view of Theorem 1.1 we may restrict our attention to the interval p , { q (say); it is enoughto show that(4.1) sup L ą l sup x Pp , { q ˇˇˇˇˇ L ÿ k “ l c k sin k α x ˇˇˇˇˇ ! b l , where b l “ sup k ě l c k k (in this paragraph the implied constants are allowed to depend on α ).Fix arbitrary x P p , { q and define L “ t p πx ´ q { α u ` , L “ t x ´ {p α ´ δ q u , where δ “ min t α { , { u . We consider three cases depending on how large L is with respect to x .a) If L ď L , then ˇˇˇˇˇ L ÿ k “ l c k sin k α x ˇˇˇˇˇ ď L ÿ k “ l c k k α x ď b l x ÿ k ď L k α ´ ! b l xL α ! b l . b) If L ă L ď L , then ˇˇˇˇˇ L ÿ k “ L c k sin k α x ˇˇˇˇˇ ! b l ` L ´ ÿ k “ L p c k ´ c k ` q| S k p x q| , MIKHAIL R. GABDULLIN where S k p x q “ ř kn “ L sin k α x . Let I m be defined as in Lemma 2.7, and let m p k q denote thelargest positive integer m with p πm { x q { α ď k . Then ď m p k q ď k α x and S k p x q “ m p k q´ ÿ m “ I m ` O ¨˚˝ ÿ p πm p k q x q { α ď n ă p π p m p k q` q x q { α ˛‹‚ . Applying Lemma 2.7, we see that ˇˇ S k p x q ˇˇ ! ÿ m ď m p k q ` x ´ { α m { α ´ ` ˘ ` max t x ´ { α m p k q { α ´ , u ! x ´ { α m p k q { α ´ ` m p k q ! k ´ α x ´ ` k α x. Therefore(4.2) ˇˇˇˇˇ L ÿ k “ L c k sin k α x ˇˇˇˇˇ ! b l ` L ÿ k “ L c k p k ´ α x ´ ` k α ´ x q ! b l ˜ ` L ÿ k “ L p k ´ α ´ x ´ ` k α ´ x q ¸ . If α P p , q , then we have L ÿ k “ L p k ´ α ´ x ´ ` k α ´ x q ! L ´ α x ´ ` x ! . If α ą , then δ “ { and we also get L ÿ k “ L p k ´ α ´ x ´ ` k α ´ x q ! L ´ α x ´ ` L α ´ x ! ` x ´p α ´ q{p α ´ { q ! . Thus in both cases from (4.2) we have ˇˇˇř Lk “ L c k sin k α x ˇˇˇ ! b l , and (4.1) follows from here andthe bound from the previous case.c) Finally, we consider the case when L ą L (in fact, this step is not needed for the case α P p , q ). We have ˇˇˇˇˇ L ÿ k “ L c k sin k α x ˇˇˇˇˇ ! b l ` L ÿ k “ L p c k ´ c k ` q| r S k p x q| , where r S k p x q “ ř kn “ L sin k α x . Lemma 2.9 implies that r S k p x q ! k ´ d p α q for some d p α q ą .Then as before ˇˇˇˇˇ L ÿ k “ L c k sin k α x ˇˇˇˇˇ ! b l ` L ÿ k “ L c k k ´ d p α q ! b l ˜ ` ÿ k “ L k ´ d p α q´ ¸ ! b l . Again, (4.1) follows from here and the bounds from the previous cases.
RIGONOMETRIC SERIES WITH NONINTEGER HARMONICS 9
Thus, (4.1) follows for any x P p , { q and L ą l . This completes the proof of Theorem1.2. 5. P ROOF OF T HEOREM
Theorem 5.1 ([Bes]) . Let p , . . . , p s be distinct primes, and b , . . . , b s be positive integers notdivisible by any of these primes. Let, further, x , . . . , x s be positive real roots of the equations x n ´ p b “ , . . . , x n s ´ p s b s “ , respectively, and P p y , ..., y s q be a nonzero polynomial with rational coefficients of degree atmost n i ´ with respect to y i for each i “ , . . . , s . Then P p x , ..., x s q ‰ . In what follows, for a number a P R we use the standard notation } a } “ min m P Z | a ´ m | . Lemma 5.2 ([Kor], Corollary 3.4) . Let δ P p , { q and let α “ p , α , . . . , α ν q P R ν ` be avector whose components are linearly independent over Q . Then there exists c “ c p α, δ q ą such that for any β , . . . , β ν P R each interval of length c contains a number x with } xα j ` β j } ă δ, j “ , . . . , ν. Fix a noninteger rational number α ą . Let n , . . . , n ν be square-free integers up to apositive integer L . We first show that the numbers n αj are linearly independent over Q . If wewrite α “ u { v , where u, v P N , u “ qv ` r , and ă r ă v , then clearly it is enough to show thatthe numbers n r { vj are linearly independent over Q . Let p , . . . , p s be primes up to L . Then anylinear combination of the numbers n r { v , . . . , n r { vν with rational coefficients can be thought of asthe value of an appropriate polynomial P p x , . . . , x s q at the point x “ p { v , . . . , x s “ p { vs ; forthis P we have deg x i P “ r ă v for each i “ , . . . , s . Applying Theorem 5.1 with b i “ and n i “ v for all i “ , . . . , s , we see that this value can be equal to zero only if all the coefficientsof our linear combination are equal to zero. Thus, the numbers n αj are linearly independent over Q .Now we show that there exists x “ x p L, α q P R such that(5.1) sin n α x ě . for all square-free n ď L . To do this, note that the numbers , n α {p π q , ..., n αν {p π q are alsolinear independent over Q (since π is transcendental, it immediately follows from the inde-pendence of n αj ); applying Lemma 5.2 with δ “ ´ and β i “ ´ { , we see that there exist x “ x p L, α q P R and integers m j with | n αj x {p π q ´ m j ´ { | ă ´ . Then | n αj x ´ πm j ´ π { | ă π { so, we have n αj x P p π { ´ π { , π { ` π { q p mod 2 π q , and (5.1) follows.Now we are ready to prove Theorem 1.3. Trivially, the condition ř k c k ă 8 is sufficientfor the uniform convergence of the series (1.1) on R , so we prove the converse. This uniform convergence is equivalent to the following: for any ε ą there exists l “ l p ε q P N such that forall L ą l we have(5.2) sup x P R ˇˇˇˇˇ L ÿ k “ l c k sin k α x ˇˇˇˇˇ ă ε, and we may suppose that l is large enough. Fix L . Using Abel’s summation, we can write(5.3) L ÿ k “ l c k sin k α x “ L ´ ÿ k “ l p c k ´ c k ` q S k p x q ´ c l S l ´ ` c L S L , where S k p x q “ ř kn “ sin n α x . It is well-known that the number of square-free positive integersup to k is π k ` O p k { q , and π “ . ... . Let A p n q be the indicator function of square-freeintegers. Then for x “ x p L, α q and all k with l ď k ď L , we have by (5.1) S k p x q “ ÿ n ď k A p n q sin n α x ` ÿ n ď k p ´ A p n qq sin n α x ě . ¨ . k ´ . k ą . k, since l , and hence k , is large enough. From here, (5.3), and monotonicity of c k , we have L ÿ k “ l c k sin k α x ą . L ´ ÿ k “ l p c k ´ c k ` q k ´ c l l ` . c L L “ . L ÿ k “ l ` c k ´ c l l. This bound implies that the series ř k c k converges, since otherwise ř Lk “ l c k sin k α x could bemade arbitrarily large by choice of L , and it would contradict (5.2). This concludes the proof.R EFERENCES [Bes] A. S. B
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OSCOW , R
USSIA , 119991
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