ON THE PROBLEM BY ERDO¨S-DE BRUIJN-KINGMAN ON REGULARITY OF RECIPROCALS FOR EXPONENTIAL SERIES 7 1 0 2 ALEXANDERGOMILKO ANDYURITOMILOV n a Abstract. Motivated by applications to renewal theory, Erdös, de J Bruijn and Kingman posed in 50th-70th a problem on regularity of re- 6 ciprocals of probability generating functions. We solve the problem in 1 the strong negative and give a numberof other related results. ] A C . 1. Introduction h t a 1.1. Motivation. The paper is devoted to solutions of certain problems m related to renewal sequences and their generating functions. Recall that if [ (a )∞ is such that a 0,k 1, and a = 1 then the sequence k k=1 k ≥ ≥ k≥1 k 1 (bk)∞k=0 given by the recurrence relation P v n 7 (1.1) b = a b , b = 1, n N, 5 n k n−k 0 ∈ 3 k=1 X 4 is called the renewal sequence associated to (a )∞ . Renewal sequences is 0 k k=1 . a classical object of studies in probability theory, in particular, in the the- 1 0 ory of Markov processes. To mention one of the probabilistic meanings of 7 (1.1), note that given a discrete Markov chain, (1.1) expresses the diagonal 1 transition probabilities (b )∞ in terms of the recurrence time probabilities : k k=0 v (a )∞ . k k=1 i X Moreover, the renewal sequences are of substantial interest in ergodic r theory. For the applications in ergodic theory one may consult e.g. the a papers [1], [2] and [16], the book [3] and the references therein. It is often convenient to study (a )∞ and (b )∞ in terms of their gen- k k=1 k k=0 erating functions F and G given by ∞ ∞ F(z) = a zk and G(z) = b zk. k k k=1 k=0 X X 1991 Mathematics Subject Classification. Primary 42A32, 42A16, 60K05; Secondary 60E10, 60J10. Key words and phrases. Renewal sequences, generating functions, Fourier coefficients, trigonometrical series, absolute convergence. This work was completed with the partial support of the NCN grant DEC- 2014/13/B/ST1/03153. 1 2 ALEXANDERGOMILKOANDYURITOMILOV The functions are defined in the open unit disc D := z C : z < 1 and { ∈ | | } connected by the relation 1 G(z) = . 1 F(z) − Beingunabletogiveanyaccountofthewidetopicofrenewalsequenceswe refertotheclassical sourcessuchasforinstance[21],[30],and[13](although the term “renewal sequence” for (b )∞ given by (1.1) is used only in [21]). n n=0 1.2. History. One of the firstand foundational results in theory of renewal sequences is the famous Erdös-Feller-Pollard theorem. To recall it we need tointroducecertain notation. Let + consist of thepower series of theform A ∞ ∞ (1.2) F(z) = a zk, a 0, a = 1 k k k ≥ k=1 k=1 X X in D. It is a complete metric space with metric induced by ℓ -norm on an 1 appropriate sequence space. We say that F + is aperiodic if F(z) = ∈ A 1,z D, implies that z = 1. ∈ Using Wiener’s theorem, it was proved in [11] that if F + is aperiodic ∈ A and additionally ∞ (1.3) µ := ka < , k ∞ k=1 X then (1.4) lim b = 1/µ. k k→∞ This is essentially the famous Erdös-Feller-Pollard theorem, one of the first and basic limit theorems in the renewal theory. The key point in [11] for showing the property (1.4) was the fact that the function (1 z)(1 F)−1 has absolutely convergent Taylor series: − − ∞ (1.5) b b < . k k+1 | − | ∞ k=1 X The theorem generated an area of research, and a huge number of its generalizations and improvements in various directions has appearedin sub- sequent years. Analytic approaches to the study of 1/(1 F) and of asymp- − totics of (b )∞ are discussed e.g. in [22, Chapter V.22] and [27, Chapter k k=1 24]. These books contain a number of related references. We mention here only the classical papers [31] and [12]. However, certain natural questions have been left open to the best of our knowledge. In particular, P. Erdös and N. de Bruijn suggested in [7, p. 164] that (1.5) is probably true for any aperiodic F + and the ∈ A assumption (1.3) is redundant. As they wrote in [7], “it seems possible that the condition (1.3) is superfluous”. Moreover, the question whether (1.5) holds for any aperiodic F satisfying (1.2) was formulated as an open ON REGULARITY OF RECIPROCALS FOR EXPONENTIAL SERIES 3 problem by J. Kingman in [21, p. 20-21, (iv)]. A recent discussion of the problemin thecontext of ergodic theory can befoundin [2]. Theanalysis of (1 z)(1 F)−1 presents certain difficulties in view of nonlinear character of − − thetransformationF (1 F)−1.While(b )∞ is given explicitly interms 7→ − k k=0 of (a )∞ , it is very difficult to study it by means of the recurrence relation k k=1 (1.1) (see e.g. [6] and [7] for such a direct approach). So most of research on analytic properties of renewal sequences concentrated on the generating functions methodology. One must note that relevant studies has been made by J. Littlewood in [25], a paper apparently overlooked by mathematical community. Being motivated by the enigmatic message from Besicovitch (see [26, p. 145]) and a question by W. Smith, Littlewood proved in [25] that for any function f given by ∞ (1.6) f(θ)= a eiλkθ, where λ R, λ 1, lim λ = , k k k ∈ ≥ k→∞ ∞ k=1 X and (a )∞ as in (1.2), one has k k=1 2θ0 dθ (1.7) lim < . θ0→0 Zθ0 |1−f(eiθ)| ∞ (Sometimes f satisfying (1.6) are called quasi-exponential series.) In particular, there is δ > 0 (depending on f) such that δ θαdθ (1.8) < 1 f(eiθ) ∞ Z0 | − | for any α > 0. The results of that type lead to a number of useful conse- quences in the study of regularity for generating functions of renewal sequences as we show in Section 5. It is natural to ask whether Littlewood’s results can essentially be im- proved. For example, the boundednessof 1 f(eiθ)/θ in the neighborhood | − | of zero would imply (1.7). Littlewood’s student H. T. Croft claimed in [8] that the latter property does not hold, in general. More precisely, if f is defined by (1.6) then for any function χ such that χ(θ) as θ 0 there ↑∞ → exist sequences (a )∞ and (λ )∞ as above, and (θ )∞ satisfying θ 0 k k=1 k k=1 k k=1 k → as k such that → ∞ (1.9) 1 f(θ ) χ(θ )θ2, k N. | − k | ≤ k k ∈ (In fact, only the case χ(θ) = θ−ǫ was discussed in [8].) This, indirectly, would solve the Erdös-de Bruijn-Kingman problem once one would arrange the integer frequencies λ above, although Croft presumably was not aware k of the problem. However, [8] contains only a hint rather than a complete argument, and it produces merely real frequencies λ rather than integer k ones as in (1.2). It is also instructive to remark that in [17] J. Hawkes constructed a lacu- nary series of the form (1.6) with θ = 2−2k2 and λ = 2π−θk2/3, k N, such k k θk ∈ 4 ALEXANDERGOMILKOANDYURITOMILOV that 1 f(θ ) k lim | − | = 0. k→∞ θk This way, Hawkes solved another Kingman’s problem formulated in [21, p. 76], which is similar (but not equivalent) to the problem mentioned above. 1.3. Results. In this paper, we answer the question by Erdös, de Bruijn and Kingman in the strong negative. Namely, we prove in Theorem 4.3 that for any positive sequence (ǫ )∞ tending to zero (subject to a technical k k=1 assumption) there exists an aperiodic F + with ∈ A ∞ (1.10) kǫ a < , k k ∞ k=1 X such that (1 z)(1 F)−1 is not even bounded in D, and thus (1.5) is not − − true. Moreover, the set of such F is dense in + (when + is considered A A as a metric space with a natural metric). Thus, the assumption (1.3) in the Erdös-Feller-Pollard theorem is best possible as far as the “smoothness” of (b )∞ is concerned. Several results of a similar nature have been obtained k k=0 as well. At the same time, we show in Appendix B that Croft’s idea can successfully be realized, and moreover it can also be realized for the integer frequencies. Our technique is based on constructing special sequences of polynomials approximating well enough a given polynomial in an appropriate norm and, asin(1.9),theconstantfunction1atasequenceofpointsfromtheunitcircle converging to 1. By means of either Baire category arguments or inductive reasoning, this then turns into the same estimates for exponential series f(θ)= F(eiθ), θ π, F +. | |≤ ∈ A It is crucial that the bounds of the type (1.9) can also be spread out to an appropriate sequence of intervals approaching 1, and thus hold on a set of sufficiently large measure. These extended bounds generalize the upper estimates from [8] and [17], and they allow us to get rid of a certain amount of regularity of (1 F)−1, e.g. with respect to the Lp-scale. − By pursuing our studies a bit further, it is natural to ask what kind regularity is possessed by (1 F)−1 without any a priori assumptions on − thesequenceofTaylorcoefficients (a )∞ ofF.Despiteenormousnumberof k k=1 papersonrenewalsequences,thequestionseemstohavenotbeenadequately addressed so far (apart probably to some extent [1], [2] and [25]) In the present paper, we make several steps in this direction. First, we extend Littlewood’s results (1.7) and (1.8) by relating the integrability of (1 F)−1 − on an interval (θ ,2θ ) (0,2) to the summability properties of the Taylor 0 0 ⊂ coefficients of F. This allows us to obtain sharp and explicit conditions for the integrability of (1 F)−1 on [ π,π] if F is aperiodic. Furthermore, − − we pursue a similar study for the “smoothed” function (1 z)(I F)−1 − − ON REGULARITY OF RECIPROCALS FOR EXPONENTIAL SERIES 5 appearing in the Erdös-de Brujin-Kingman problem. We show that for F as in (1.2), satisfying ∞ (1.11) kνa < k ∞ k=1 X for some ν (0,1), one has ∈ (1 z)(1 F)−1 L [ π,π]. 1+1/(1−ν) − − ∈ − On the other hand, for each p (2+(1 ν)−1, ) we construct a function ∈ − ∞ F of the form (1.2) satisfying (1.11) but at the same time violating p (1 z)(1 F )−1 Lp[ π,π]. p − − 6∈ − Remark that while (1.5) is not, in general, true for F + (as we show ∈ A in this paper) we prove that nevertheless a weaker property holds: ∞ (b b )2 < . k k+1 − ∞ k=0 X Thissimpleresulthasprobablybeenoverlooked intheliterature. Moreover, we show that, in general, (1 z)(1 F)−1 Lp[ π,π] if p (3, ). The − − 6∈ − ∈ ∞ problem what happens if p (2,3] remains, unfortunately, open. ∈ 2. Preliminaries and notations For w = (w )∞ (1, ) we denote by (ω) a Banach space k k=1 ⊂ ∞ A ∞ ∞ f(θ)= a eikθ : w a < , θ π , k k k | | ∞ | |≤ ( ) k=1 k=1 X X with the norm ∞ f = w a , f (w). A(w) k k k k | | ∈ A k=1 X Its subset +(w) given by A ∞ ∞ +(w) = a eikθ (ω) : a 0, a = 1 k k k A ∈A ≥ ( ) k=1 k=1 X X is a complete metric space with the metric ρ(, ) inherited from (w). A(w) · · A Note that (2.1) f(θ) g(θ) f g , f,g +(w), θ π. A(w) | − | ≤ k − k ∈ A | |≤ We will often be using a more intuitive notation f f instead 1 2 A(w) k − k of ρ(f ,f ) whenever it is defined correctly. If w = kν,ν [0,1), for 1 2 A(w) k ∈ k N,thenwewillwrite (ν)insteadof (w)slightlyabusingournotation. ∈ A A We will also write + (respectively ) instead of +( 1 ) (respectively A A A { } ( 1 )). A { } 6 ALEXANDERGOMILKOANDYURITOMILOV In the sequel, we identify absolutely convergent power series F on D with theirboundaryvaluesonT,andtheboundaryvalueswiththecorresponding 2π-periodic functions f, so that ∞ (2.2) f(θ):= F(eiθ)= a eikθ, θ π. k | |≤ k=1 X Let us recall that f + is aperiodic if and only if the greatest common factor of n N : a >∈ A0 is 1, see e.g. [30, p. 85] or [13, Vol II, p. 500- n { ∈ } 501]. In particular, any function f + of the form (2.2) with a = 0, is 1 ∈ A 6 aperiodic. Observe that the set of aperiodic polynomials from +(w) is dense in A +(w). Indeed, let f +(w) be given by A ∈ A ∞ (2.3) f(θ)= a eikθ, a = 0. k m 6 k=m X Let us define for n m+1 the family of aperiodic polynomials ≥ n a 1 P (θ)= meiθ +a 1 eimθ + a eikθ /d +, n m k n n − n ∈ A ! (cid:18) (cid:19) k=m+1 X where n d = a 1, n . n m → → ∞ k=m X Then f P (1/d 1) f +d−1 f d P k − nkA(w) ≤ n − k kA(w) n k − n nkA(w) (w +w )a 1 m m (1/d 1) f + n A(w) ≤ − k k nd n ∞ 1 + w a 0, n . k k d → → ∞ n k=n+1 X The next simple proposition will be useful for the sequel. It is probably known, but we were not able to find an appropriate reference. Proposition2.1. Thesetofaperiodic functionsin +(w)isopenin +(w). A A Proof. Let (f )∞ +(w) be a sequence of non-aperiodic functions such n n=1 ⊂ A that (2.4) lim f f = 0. 0 n A(w) n→∞ k − k Note that for every n N there exists θ [π/2,π] such that f (θ ) = 1. If n n n ∈ ∈ θ is any limit point of (θ )∞ then θ [π/2,π], and from (2.4), (2.1) and 0 n n=1 0 ∈ thecontinuity of f itfollows thatf (θ )= 1. Therefore, f isnotaperiodic, 0 0 0 0 and the set of non-aperiodic functions is closed in +(w). (cid:3) A ON REGULARITY OF RECIPROCALS FOR EXPONENTIAL SERIES 7 Remark 2.2. By Proposition 2.1 the set of aperiodic functions in +(w) is A open in +(w). Since that set is also dense in +(w) as we showed above, A A the set of aperiodic functions in +(w) is residual. A Finally, we will fix some standard notation for the rest of the paper. For anymeasurablesetE R(orE T)weletmeas(E)standforitsLebesgue ⊂ ⊂ measure. Ausualmaxnorminthespaceof2π-periodiccontinuousfunctions on [ π,π] will be denoted by . For an exponential polynomial P + ∞ − k·k ∈ A its degree will be denoted by degP. Sometimes, to simplify the exposition, the constants will change from line to line, although in several places we will give the precise values of constants to underline their (in-)dependence on parameters. 3. Auxiliary estimates of the exponential polynomials In this section, we first obtain the lower estimates for the size of approxi- mations of the constant function 1 by exponential polynomials. Then in the next section these estimates will be extended to exponential series by either Baire category arguments or inductive constructions. We start with following technical lemma. Lemma 3.1. For λ (0,1] and γ (0,1] define ∈ ∈ sin(λ/2)cos((γ +λ)/2) d = . λ,γ sin(γ/2) Then λ λ λ(γ +λ) (3.1) d and (1 eiλ)+d (1 e−iγ) . λ,γ λ,γ 4γ ≤ ≤ γ | − − | ≤ 2 Since the proof of Lemma 3.1 is based on simple computations with trigonometrical functions, it will be postponed to Appendix A. Thenextcorollary givesarecipeforconstructingexponentialpolynomials (having, in general, non-integer frequencies) with control of their size at a fixed point and of their variation on the unit circle. Corollary 3.2. Let P(θ) = n a eikθ +. For all θ (0,1/n] and k=1 k ∈ A 0 ∈ γ (0,1] there exists d θ0,nθ0 such that if ∈ ∈ 4γ Pγ h n i (3.2) Pd,γ(θ):= ak eikθ + d ei(2π−γ)θθ0, 1+d 1+d k=1 X then 2π (3.3) 1 P (θ ) 2nθ γ and P′ n 1+ . | − d,γ 0 | ≤ 0 k d,γk∞ ≤ γ (cid:18) (cid:19) Proof. Let θ (0,1/n], and γ (0,1] be fixed. Set 0 ∈ ∈ n d := a d , k kθ0,γ k=1 X 8 ALEXANDERGOMILKOANDYURITOMILOV where d ,1 k n, are given by Lemma 3.1. Then, by Lemma 3.1, kθ0,γ ≤ ≤ n n θ 1 1 nθ 0 0 (3.4) ka θ d ka θ . k 0 k 0 4γ ≤ 4γ ≤ ≤ γ ≤ γ k=1 k=1 X X Note that n (1+d)(1 P (θ )) = a [(1 eikθ0)+(1 e−iγ)]. d,γ 0 k − − − k=1 X So using (3.1) and (3.4), we obtain that n (1+d)1 P (θ ) a (1 eikθ0)+d (1 e−iγ) | − d,γ 0 | ≤ k| − kθ0,γ − | k=1 X n 1 ka θ (γ+kθ ) k 0 0 ≤2 k=1 X n nθ γ kθ 0 0 a 1+ k ≤ 2 γ k=1 (cid:18) (cid:19) X 2nθ γ(1+d), 0 ≤ hence the first estimate in (3.3) holds. Finally, by (3.4), n 2π γ 2πnθ 2π P′ ka +d − n+ 0 = n 1+ , k d,γk∞ ≤ k θ ≤ θ γ γ k=1 0 0 (cid:18) (cid:19) X i.e. the second estimate in (3.3) is true. (cid:3) Nowweareabletoshowthatforanypolynomialfrom + thereisanother A polynomial close to it in an appropriate weighted norm and close to the function 1 on a sequence of points of T going to 1. Theorem 3.3. Let (ǫ )∞ be a positive sequence such that k k=1 lim ǫ = 0 and kǫ 1, k N, k→∞ k k ≥ ∈ and let w˜ = (kǫ )∞ . Then for every polynomial P + there exist a se- k k=1 ∈ A quence(θ )∞ decreasing tozeroandasequenceofpolynomials (Q )∞ m m=1 m m=1 ⊂ + satisfying A 1 Q (θ ) m m lim | − | =0 and lim P Q = 0. m A(w˜) m→∞ θm m→∞ k − k Proof. LetapolynomialP befixed,andletdegP =n.Definee := sup ǫ : n k { 1 k n , and choose a subsequence (ǫ ) such that ≤ ≤ } sm m≥1 lim ǫ = 0. m→∞ sm ON REGULARITY OF RECIPROCALS FOR EXPONENTIAL SERIES 9 Fix an integer m > 2πn such that 1/2 ne n (3.5) γ := ǫ + 1 m sm s ≤ (cid:18) m (cid:19) and put 2π γ m θ := − . m s m Using Corollary 3.2 with θ = θ and γ = γ we conclude that there 0 m m exist d , 0 < d nθ /γ , and a polynomial Q := P +, m m ≤ m m m dm,γm ∈ A degQ = s > n, defined by m m n Q (θ)= b eikθ +b eismθ, m k sm k=1 X such that 1 Q (θ ) 2d 2nθ m m m m | − | 2nγ and P Q = . m m A θ ≤ k − k 1+d ≤ γ m m m Hence (3.5) and the latter inequality imply that 2nθ P Q max(ne ,s ǫ ) P Q s γ2 m 4πnγ . k − mkA(w˜) ≤ n m sm k − mkA ≤ m m γ ≤ m m Since γ 0 as m , the statement follows. (cid:3) m → → ∞ Remark 3.4. Here and the sequel, the assumption kǫ 1,k N, is of k ≥ ∈ purely technical nature and has been made to simplify our exposition. Recall from Section 2 that the set of polynomials in +(w) is dense in A +(w) for any weight w, thus Theorem 3.3 implies the following statement. A Corollary 3.5. Let (ǫ )∞ (0, ) satisfy lim ǫ = 0 and kǫ 1, k N, and let w˜ = (kkkǫ=)1∞⊂. Th∞en for every kf→∞ k(w˜) there exkis≥ts a ∈ k k=1 ∈ A sequence of polynomials (Q )∞ + such that m m=1 ∈ A 1 Q (θ) m lim inf | − | = 0 and lim f Q =0. m A(w˜) m→∞ θ∈(0,1/m] θ m→∞ k − k The next result is our basic statement allowing one to spread out the upper estimates for 1 Q proved in Theorem 3.3 from the sequence m | − | (θ )∞ to a larger set containing it. The result will help us to provide m m=1 counterexamples on Lp-integrability of (1 z)(1 F)−1. − − Theorem 3.6. Let ψ : (0,1] (0, ) and χ : (0,1] (0, ) be continuous 7→ ∞ 7→ ∞ functions satisfying χ(θ) (3.6) lim ψ(θ) = 0 and lim = 0. θ→0 θ→0 ψ(θ) Then for every polynomial P + there exist a sequence (θ )∞ decreas- ∈ A m m=1 ing to zero and a sequence of polynomials (Q )∞ + such that for all m N: m m=1 ⊂ A ∈ 10 ALEXANDERGOMILKOANDYURITOMILOV (i) mθ 2π, m ≤ (ii) 1 Q (θ ) 2ψ(θ )θ , m m m m | − | ≤ (iii) P Q 2θm . k − mkA ≤ χ(θm) Moreover, for each m N and for each θ such that θ θ ψ(θ )χ(θ )θ m m m m ∈ | − | ≤ one has 1 Q (θ) 10ψ(θ )θ . m m m | − |≤ Proof. Let a polynomial P be fixed, and let degP = n. Making use of (3.6), choose Θ (0,1] in such a way that 0 ∈ χ1/2(θ) (3.7) χ(θ)ψ(θ)+n +θ 1, θ (0,Θ ]. 0 ψ1/2(θ) !≤ ∈ Define 2π γ(θ) τ(θ):= − , γ(θ):= ψ1/2(θ)χ1/2(θ), θ (0,Θ ], 0 θ ∈ and note that, in particular, nθ 1, and γ(θ) 1 for all θ (0,Θ ]. 0 ≤ ≤ ∈ Moreover, if θ (0,Θ ], then from (3.7) it follows that 0 ∈ ψ1/2(θ)γ(θ) n ψ1/2(θ) 1 (3.8) nγ(θ) = ψ(θ), = . ≤ χ1/2(θ) γ(θ) ≤ γ(θ)χ1/2(θ) χ(θ) Since γ is continuous on (0,Θ ] and lim τ(θ) = + , there exists a 0 θ→0 ∞ sequence (θ ) [0,Θ ] satisfying m m≥m0 ⊂ 0 τ(θ ) = m, m m . m 0 ≥ Moreover, as lim γ(θ) = 0, we may also assume that mθ 2π. θ→0 m ≤ Next, we fix m max(m ,n), set θ = θ and γ = γ(θ ), and apply 0 0 m m ≥ Corollary 3.2 to the polynomial P. Taking into account (3.8), we infer that there exist d > 0 and a polynomial Q = P + such that m m dm,γ(θm) ∈ A 1 Q (θ ) 2nγ(θ )θ 2ψ(θ )θ , m m m m m m | − | ≤ ≤ and 2nθ 2θ m m P Q . m A k − k ≤ γ(θ ) ≤ χ(θ ) m m Moreover, by (3.3) and (3.8), 2π γ(θ ) 2π (1+2π) (3.9) Q′ n 1+ m 1+ . k mk∞ ≤ γ(θ ) ≤ χ(θ ) γ(θ ) ≤ χ(θ ) (cid:18) m (cid:19) m (cid:18) m (cid:19) m Using θ (3.10) 1 Q (θ) 1 Q (θ ) + Q′ (s) ds, | − m |≤ | − m m | | m | Zθm and (3.9), we conclude that if θ θ ψ(θ )χ(θ )θ , then m m m m | − | ≤ 1+2π 1 Q (θ) 2ψ(θ )θ + θ θ 10ψ(θ )θ . m m m m m m | − |≤ χ(θ )| − | ≤ m (cid:3)

On the problem by Erd\"os-de Bruijn-Kingman on regularity of reciprocals for exponential series

Similar On the problem by Erd\"os-de Bruijn-Kingman on regularity of reciprocals for exponential series