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

TL;DR

This paper investigates the boundedness and integrability properties of reciprocals of probability generating functions, providing stronger counterexamples and systematic analysis, with implications for renewal theory and continuous-time processes.

Contribution

It offers new, stronger counterexamples to a classical problem and systematically studies $L^p$-integrability of reciprocals under various conditions.

Findings

Boundedness of reciprocals generally fails.

Reciprocals exhibit certain $L^p$-integrability properties.

Results extend to continuous-time settings.

Abstract

Motivated by applications to renewal theory, Erd\H{o}s, de Bruijn and Kingman posed a problem on boundedness of reciprocals $(1-z)/(1-F(z))$ in the unit disc for probability generating functions $F(z)$. It was solved by Ibragimov in $1975$ by constructing a counterexample. In this paper, we provide much stronger counterexamples showing that the problem does not allow for a positive answer even under rather restrictive additional assumptions. Moreover, we pursue a systematic study of $L^p$-integrabilty properties for the reciprocals. In particular, we show that while the boundedness of $(1-z)/(1-F(z))$ fails in general, the reciprocals do possess certain $L^p$-integrability properties under mild conditions on $F$. We also study the same circle of problems in the continuous-time setting.

Equations

$$b_n=\sum _{k=1}^n{a}_k{b}_{n-k},b_0=1,n\in \mathbb{N},$$

$$F(z)=\sum _{k=1}^{\infty }{a}_k{z}^k\text{and}G(z)=\sum _{k=0}^{\infty }{b}_k{z}^k.$$

$$G(z)=\frac{1}{1-F(z)}.$$

$$F(z)=\sum _{k=1}^{\infty }{a}_k{z}^k,a_k\ge 0,\sum _{k=1}^{\infty }{a}_k=1$$

$$\mu :=\sum _{k=1}^{\infty }k{a}_k<\infty ,$$

$$\underset{k\to \infty }{\lim }{b}_k=1/\mu .$$

$$\sum _{k=1}^{\infty }|b_k-b_{k+1}|<\infty .$$

$$f(\theta )=\sum _{k=1}^{\infty }{a}_k{e}^{i{\lambda }_k\theta },\text{where}{\lambda }_k\in \mathbb{R},{\lambda }_k\ge 1,\underset{k\to \infty }{\lim }{\lambda }_k=\infty ,$$

$$\overline{\underset{{\theta }_0\to 0}{\lim }}{\int }_{{\theta }_0}^{2{\theta }_0}\frac{d\theta }{|1-f(e^{i\theta })|}<\infty .$$

$${\int }_0^{\delta }\frac{{\theta }^{\alpha }d\theta }{|1-f(e^{i\theta })|}<\infty$$

$$|1-f({\theta }_k)|\le \chi ({\theta }_k){\theta }_k^2,k\in \mathbb{N}.$$

$$\underset{k\to \infty }{\lim }\frac{|1-f({\theta }_k)|}{{\theta }_k}=0.$$

$$\sum _{k=1}^{\infty }k{ϵ}_k{a}_k<\infty ,$$

$$f(\theta )=F(e^{i\theta }),|\theta |\le \pi ,F\in {\mathcal{A}}^{+}.$$

$$\sum _{k=1}^{\infty }{k}^{\nu }{a}_k<\infty$$

$$(1-z)(1-F{)}^{-1}\in L^{1+\frac{1}{1-\nu }}(\mathbb{T}).$$

$$(1-z)(1-F_p{)}^{-1}\notin L^p(\mathbb{T}).$$

$$\sum _{k=0}^{\infty }(b_k-b_{k+1}{)}^2<\infty .$$

$$\left\{f(\theta )=\sum _{k=1}^{\infty }{a}_k{e}^{ik\theta }:\sum _{k=1}^{\infty }{w}_k|a_k|<\infty ,|\theta |\le \pi \right\},$$

$$\Vert f{\Vert }_{\mathcal{A}(w)}=\sum _{k=1}^{\infty }{w}_k|a_k|,f\in \mathcal{A}(w).$$

$${\mathcal{A}}^{+}(w)=\left\{\sum _{k=1}^{\infty }{a}_k{e}^{ik\theta }\in \mathcal{A}(\omega ):a_k\ge 0,\sum _{k=1}^{\infty }{a}_k=1\right\}$$

$$|f(\theta )-g(\theta )|\le \Vert f-g{\Vert }_{\mathcal{A}(w)},f,g\in {\mathcal{A}}^{+}(w),|\theta |\le \pi .$$

$$f(\theta ):=F(e^{i\theta })=\sum _{k=1}^{\infty }{a}_k{e}^{ik\theta },|\theta |\le \pi .$$

$$f(\theta )=\sum _{k=m}^{\infty }{a}_k{e}^{ik\theta },a_m\ne 0.$$

$$P_n(\theta )=\left(\frac{a_m}{n}{e}^{i\theta }+a_m\left(1-\frac{1}{n}\right)e^{im\theta }+\sum _{k=m+1}^n{a}_k{e}^{ik\theta }\right)/d_n\in {\mathcal{A}}^{+},$$

$$d_n=\sum _{k=m}^n{a}_k\to 1,n\to \infty .$$

$$\Vert f-P_n{\Vert }_{\mathcal{A}(w)}\le$$

$$\le$$

$$+$$

$$\underset{n\to \infty }{\lim }\Vert f_0-f_n{\Vert }_{\mathcal{A}(w)}=0$$

$$d_{\lambda ,\gamma }=\frac{\sin (\lambda /2)\cos ((\gamma +\lambda )/2)}{\sin (\gamma /2)}.$$

Full text

1. Introduction

1.1. Motivation

The paper addresses several notorious problems related to renewal sequences and their generating functions. Recall that if $(a_{k})_{k=1}^{\infty}$ is such that $a_{k}\geq 0,k\geq 1,$ and $\sum_{k\geq 1}a_{k}=1,$ then the sequence $(b_{k})_{k=0}^{\infty}$ given by the recurrence relation

[TABLE]

is called the renewal sequence associated to $(a_{k})_{k=1}^{\infty}.$ Renewal sequences are a classical subject of studies in probability theory, in particular, in the theory of Markov processes. To mention one of the probabilistic meanings of (1.1), let $\{X_{n}:n\geq 0\}$ be a recurrent Markov chain with the state space $\mathbb{N}\cup\{0\}$ and $P(X_{0}=0)=1$. If $T:=\inf\{n\geq 1:X_{n}=0\}$ denotes the time of first return to the origin, and $a_{n}:=P(T=n),n\in\mathbb{N},$ then $b_{n}=P(X_{n}=0|X_{0}=0),n\in\mathbb{N}.$ For a thorough discussion of probabilistic background for (1.1), see e.g. [13, Vol 1, Ch. XIII] and [23, Ch. 1].

Moreover, renewal sequences are also of substantial interest in ergodic theory. For applications in ergodic theory one may consult e.g. the papers [1], [2] and [16], the book [3] and the references therein.

It is often convenient to study $(a_{k})_{k=1}^{\infty}$ and $(b_{k})_{k=0}^{\infty}$ in terms of their generating functions $F$ and $G,$ given by

[TABLE]

The functions are defined on the open unit disc $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ and connected by the relation

[TABLE]

Being unable to give any account of the wide topic of renewal sequences we refer to the classical sources such as for instance [23], [20], [32], and [13] (although the term “renewal sequence” for $(b_{n})_{n=0}^{\infty}$ given by (1.1) is used only in [23] and [20]).

1.2. History

One of the first and foundational results in theory of renewal sequences is the famous Erdős-Feller-Pollard theorem. To recall it we need to introduce certain notation. Let $\mathcal{A}^{+}$ consist of the power series of the form

[TABLE]

in $\overline{\mathbb{D}}.$ It is a complete metric space with metric induced by $\ell_{1}$-norm on an appropriate sequence space. We say that $F\in\mathcal{A}^{+}$ is aperiodic if $F(z)=1,z\in\overline{\mathbb{D}},$ implies that $z=1.$ Clearly, if $F$ is aperiodic then $1/(1-F)$ is analytic in $\mathbb{D}$ and continuous in ${\mathbb{D}}\setminus\{1\}.$

Using Wiener’s theorem, it was proved in [11] that if $F\in\mathcal{A}^{+}$ is aperiodic and additionally

[TABLE]

then

[TABLE]

This is essentially the famous Erdős-Feller-Pollard theorem, one of the first and basic limit theorems in 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:

[TABLE]

The theorem generated an area of research, and a huge number of its generalizations and improvements in various directions has appeared in subsequent years. Analytic approaches to the study of $1/(1-F)$ and of asymptotics of $(b_{k})_{k=1}^{\infty}$ are discussed e.g. in [24, Chapter V.22] and [29, Chapter 24]. These books contain a number of related references. We mention here only the classical papers [33] and [12].

However, certain natural questions have escaped a thorough study. In particular, P. Erdős and N. de Bruijn suggested in [7, p. 164] that (1.5) is probably true for any aperiodic $F\in\mathcal{A}^{+}$ and the 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 problem by J. Kingman in [23, p. 20-21, (iv)]. A recent discussion of the problem in the context of ergodic theory can be found in [2]. The analysis of $(1-z)(1-F)^{-1}$ presents certain difficulties in view of nonlinear character of the transformation $F\mapsto(1-F)^{-1}.$ While $(b_{k})_{k=0}^{\infty}$ is given explicitly in terms of $(a_{k})_{k=1}^{\infty},$ it is very difficult to study it by means of the recurrence relation (1.1) (see e.g. [6] and [7] for such a direct approach). So most of the research on analytic properties of renewal sequences concentrated on the generating-function methodology.

One must note that relevant studies have been made by J. Littlewood in [27], a paper apparently overlooked by mathematical community. Being motivated by the enigmatic message from Besicovitch (see [28, p. 145]) and a question by W. L. Smith, Littlewood proved in [27] that for any function $f$ given by

[TABLE]

and $(a_{k})_{k=1}^{\infty}$ as in (1.2), one has

[TABLE]

(Sometimes $f$ satisfying (1.6) are called quasi-exponential series.) In particular, there is $\delta>0$ (depending on $f$) such that

[TABLE]

for any $\alpha>0.$ Results of that type lead to a number of useful consequences 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 be essentially improved. For example, boundedness from below of $|1-f(e^{i\theta})|/\theta$ 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 $\chi$ such that $\chi(\theta)\uparrow\infty$ as $\theta\to 0$ there exist sequences $(a_{k})_{k=1}^{\infty}$ and $(\lambda_{k})_{k=1}^{\infty}$ as above, and $(\theta_{k})_{k=1}^{\infty}$ satisfying $\theta_{k}\to 0$ as $k\to\infty$ such that

[TABLE]

(In fact, only the case $\chi(\theta)=\theta^{-\epsilon}$ was discussed in [8].) This, indirectly, would solve the Erdős-de Bruijn-Kingman problem once one would arrange the integer frequencies $\lambda_{k}$ above, although Croft presumably was not aware of the problem. However, [8] contains only a hint rather than a complete argument, and it produces merely real frequencies $\lambda_{k}$ rather than integer ones as in (1.2).

The Erdős-de Bruijn-Kingman problem was settled in the negative by I. A. Ibragimov in [19], who constructed an $F$ such that $(1-z)/(1-F(z))$ is unbounded in $\mathbb{D}.$ However, the size of the gap between the Erdős-Feller-Pollard condition (1.3) and the situation with no a priori assumption, i.e. $\sum_{k\geq 1}a_{k}=1,$ remained completely unclear. In this paper, we show that (1.5) does not in general hold under essentially any summability assumptions weaker than (1.3), and moreover (1.5) fails for generic probabilities $(a_{k})_{k=1}^{\infty}.$

It is also instructive to remark that in [17] J. Hawkes constructed a lacunary series of the form (1.6) with $\theta_{k}=2^{-2^{k^{2}}}$ and $\lambda_{k}=\frac{2\pi-\theta_{k}^{2/3}}{\theta_{k}},\,k\in\mathbb{N},$ such that

[TABLE]

This way, Hawkes solved another Kingman problem formulated in [23, p. 76], which is similar (but not equivalent) to the problem mentioned above. However, the approaches of [17] and [23] are quite close to each other.

1.3. Results

In this paper, we revisit the problem posed by Erdős, de Bruijn and Kingman, and provide counterexamples that can be considered as, in a sense, best possible. Namely, by methods very different from those of [19], we prove in Theorem 4.3 that for any positive sequence $(\epsilon_{k})_{k=1}^{\infty}$ tending to zero (subject to a technical assumption) there exists an aperiodic $F\in\mathcal{A}^{+}$ with

[TABLE]

such that $(1-z)(1-F)^{-1}$ is not even bounded in $\mathbb{D}$, and thus (1.5) is not true. Moreover, the set of such $F$ is dense in $\mathcal{A}^{+}$ (when $\mathcal{A}^{+}$ is considered as a metric space with a natural metric). Thus, the assumption (1.3) in the Erdős-Feller-Pollard theorem is optimal as far as the “smoothness” of $(b_{k})_{k=0}^{\infty}$ is concerned. Several results of a similar nature have been obtained 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 integer frequencies.

Our technique is based on constructing special sequences of polynomials approximating a given polynomial well enough in an appropriate norm and, as in (1.9), the constant function $1$ at a sequence of points from the unit circle converging to $1.$ By means of either Baire-category arguments or inductive reasoning, this then turns into the same estimates for exponential series:

[TABLE]

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], [19] 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 $L^{p}$-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 the sequence of Taylor coefficients $(a_{k})_{k=1}^{\infty}$ of $F.$ Despite the enormous number of papers on renewal sequences, the question seems to have not been adequately addressed so far (apart perhaps to some extent [1], [2] and [27]) 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 $(\theta_{0},2\theta_{0})\subset(0,2)$ to the summability properties of the Taylor coefficients of $F.$ This allows us to obtain sharp and explicit conditions for the integrability of $(1-F)^{-1}$ on the unit circle $\mathbb{T}$ if $F$ is aperiodic. Furthermore, we pursue a similar study for the “smoothed” function $(1-z)(I-F)^{-1}$ appearing in the Erdős-de Brujin-Kingman problem. We show that for $F$ as in (1.2), satisfying

[TABLE]

for some $\nu\in(0,1),$ one has

[TABLE]

On the other hand, for each $p\in(2+(1-\nu)^{-1},\infty)$ we construct a function $F_{p}$ of the form (1.2) satisfying (1.11) but at the same time violating

[TABLE]

Remark that while (1.5) is not, in general, true for $F\in\mathcal{A}^{+}$ (as we show in this paper), we prove that nevertheless a weaker property holds:

[TABLE]

This simple result has probably been overlooked in the literature. Moreover, we show that, in general, $(1-z)(1-F)^{-1}\not\in L^{p}(\mathbb{T})$ if $p\in(3,\infty).$ The problem what happens if $p\in(2,3]$ remains, unfortunately, open.

We finish the paper with remarks on a continuous analogue of the Erdős-de Bruijn-Kingman problem.

2. Preliminaries and notations

For $w=(w_{k})_{k=1}^{\infty}\subset[1,\infty)$ we denote by $\mathcal{A}(\omega)$ a Banach space

[TABLE]

with the norm

[TABLE]

Its subset $\mathcal{A}^{+}(w)$ given by

[TABLE]

is a complete metric space with the metric $\rho(\cdot,\cdot)_{\mathcal{A}(w)}$ inherited from $\mathcal{A}(w).$

Note that

[TABLE]

We will often use a more intuitive notation $\|f_{1}-f_{2}\|_{\mathcal{A}(w)}$ instead of $\rho(f_{1},f_{2})_{\mathcal{A}(w)}$ whenever it is defined correctly. If $w_{k}=k^{\nu},\nu\in[0,1),$ for $k\in\mathbb{N},$ then we will write $\mathcal{A}(\nu)$ instead of $\mathcal{A}(w)$ slightly abusing our notation. The same will apply to the spaces $\mathcal{A}^{+}(w).$ We will also write $\mathcal{A}^{+}$ (respectively $\mathcal{A}$) instead of $\mathcal{A}^{+}(\{1\})$ (respectively $\mathcal{A}(\{1\})$). Clearly, $\mathcal{A}$ is embedded contractively into any $A(w)$ with $w$ as above. It will sometimes be convenient to consider functions from $A(w)$ extended periodically to the whole real line.

In the sequel, we identify absolutely convergent power series $F(z)=\sum_{k=1}^{n}a_{k}z^{k}$ on $\overline{\mathbb{D}}$ with their boundary values on $\mathbb{T},$ and the boundary values with the corresponding $2\pi$-periodic functions $f,$ so that

[TABLE]

In this way, our notation $\mathcal{A}^{+}$ agrees with the same notation used in the introduction for power series. A function $f\in\mathcal{A}^{+}$ of the form $f=\sum_{k=1}^{n}a_{k}e^{ik\theta},n\in\mathbb{N},$ will be called (exponential) polynomial. As usual, its degree ${\rm deg}\,f$ is defined as ${\rm deg}\,f:=\max\{k\in\mathbb{N}:a_{k}\neq 0\}.$ Clearly the set of polynomials is contained in any $\mathcal{A}(w).$

Following the definition of aperiodic $F$ from the introduction, $f\in\mathcal{A}^{+}$ is said to be aperiodic if $f(\theta)=1$ for some $\theta\in[-\pi,\pi],$ implies $\theta=0.$ Let us recall that $f(\theta)=\sum_{k=1}^{\infty}a_{k}e^{ik\theta}\in\mathcal{A}^{+}$ is aperiodic if and only if the greatest common divisor of $\{k\in\mathbb{N}:a_{k}>0\}$ is $1.$ The argument for the “only if” part of this equivalence can be found e.g. in [23, p. 12] or [24, p. 272-273], while the other part is obvious. The equivalence, in particular, implies that if $a_{1}\not=0,$ then $f$ is aperiodic. Moreover, if $f$ is not aperiodic, then there exists $\theta\in[\pi/2,\pi]$ such that $f(\theta)=1.$

Observe that the set of polynomials in $\mathcal{A}^{+}(w)$ is dense in $\mathcal{A}^{+}(w)$ for any weight $w.$ Indeed, let $f\in\mathcal{A}^{+}(w)$ be given by

[TABLE]

Let us define for $n\geq m+1$ the family of aperiodic polynomials

[TABLE]

where

[TABLE]

Then

[TABLE]

The next simple proposition will be useful for the sequel. It is probably known, but we were not able to find an appropriate reference.

Proposition 2.1**.**

The set of aperiodic functions in $\mathcal{A}^{+}(w)$ is open in $\mathcal{A}^{+}(w)$.

Proof.

Let $(f_{n})_{n=1}^{\infty}\subset\mathcal{A}^{+}(w)$ be a sequence of non-aperiodic functions such that

[TABLE]

for some $f_{0}\in A^{+}(w).$ Note that for every $n\in\mathbb{\mathbb{N}}$ there exists $\theta_{n}\in[\pi/2,\pi]$ such that $f_{n}(\theta_{n})=1.$ If $\theta_{0}$ is any limit point of $(\theta_{n})_{n=1}^{\infty},$ then $\theta_{0}\in[\pi/2,\pi],$ and from (2.4), (2.1) and the continuity of $f_{0}$ it follows that $f_{0}(\theta_{0})=1$. Therefore, $f_{0}$ is not aperiodic, and the set of non-aperiodic functions is closed in $\mathcal{A}^{+}(w)$. ∎

Remark 2.2*.*

By Proposition 2.1 the set of aperiodic functions in $\mathcal{A}^{+}(w)$ is open in $\mathcal{A}^{+}(w).$ Since that set is also dense in $\mathcal{A}^{+}(w)$ as we showed above, the set of aperiodic functions in $\mathcal{A}^{+}(w)$ is residual, i.e. it is the complement of a set of first category in $\mathcal{A}^{+}(w).$

Finally, we will fix some standard notation for the rest of the paper. For any measurable set $E\subset\mathbb{R}$ (or $E\subset\mathbb{T}$) we let ${\rm meas}\,(E)$ stand for its Lebesgue measure. The usual max norm in the space of $2\pi$-periodic continuous functions on $[-\pi,\pi]$ will be denoted by $\|\cdot\|_{\infty}.$ 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 for the exponential polynomials

In this section, we first obtain lower estimates for the size of approximations 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 the following technical lemma.

Lemma 3.1**.**

For $\lambda,\gamma\in(0,1],$ define

[TABLE]

Then

[TABLE]

Since the proof of Lemma 3.1 is based on simple computations with trigonometrical functions, it will be postponed to Appendix A.

The next corollary gives a recipe for constructing polynomials (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 $P(\theta)=\sum_{k=1}^{n}a_{k}e^{ik\theta}\in\mathcal{A}^{+}.$ For all $\theta_{0}\in(0,1/n]$ and $\gamma\in(0,1],$ there exists $d\in\left[\frac{\theta_{0}}{4\gamma},\frac{n\theta_{0}}{\gamma}\right]$ such that if

[TABLE]

then

[TABLE]

Proof.

Let $\theta_{0}\in(0,1/n],$ and $\gamma\in(0,1]$ be fixed. Set

[TABLE]

where $d_{k\theta_{0},\gamma},1\leq k\leq n,$ are given by Lemma 3.1. Then, by Lemma 3.1,

[TABLE]

Note that

[TABLE]

So using (3.1) and (3.4), we obtain that

[TABLE]

hence the first estimate in (3.3) holds.

Finally, by (3.4),

[TABLE]

i.e. the second estimate in (3.3) is true. ∎

Now we are able to show that for any polynomial from $\mathcal{A}^{+}$ there is another polynomial close to it in an appropriate weighted norm and close to the function $1$ on a sequence $(e^{i\theta_{m}})_{m=1}^{\infty}\subset\mathbb{T}$ going to $1\in\mathbb{T}.$

Theorem 3.3**.**

Let $(\epsilon_{k})_{k=1}^{\infty}$ be a positive sequence such that

[TABLE]

and let $\tilde{w}=(k\epsilon_{k})_{k=1}^{\infty}.$ Then for every polynomial $P(\theta)=\sum_{k=1}^{n}a_{k}e^{ik\theta}\in\mathcal{A}^{+}$ there exist a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1/n]$ decreasing to zero and a sequence of polynomials $(Q_{m})_{m=1}^{\infty}\subset\mathcal{A}^{+}$ satisfying

[TABLE]

Proof.

Let a polynomial $P$ be fixed, and let ${\rm deg}\,P=n.$ Define $e_{n}:=\sup\{\epsilon_{k}:1\leq k\leq n\},$ and choose a subsequence $(\epsilon_{s_{m}})_{m\geq 1}$ such that

[TABLE]

Fix an integer $m>2\pi n$ such that

[TABLE]

and put

[TABLE]

Using Corollary 3.2 with $\theta_{0}=\theta_{m}$ and $\gamma=\gamma_{m}$ we conclude that there exist $d_{m}$, $0<d_{m}\leq n\theta_{m}/\gamma_{m},$ and a polynomial $Q_{m}:=P_{d_{m},\gamma_{m}}\in\mathcal{A}^{+},$ ${\rm deg}\,Q_{m}=s_{m}>n,$ given by

[TABLE]

such that

[TABLE]

Hence (3.5) and the latter inequality imply that

[TABLE]

Since $\gamma_{m}\to 0$ as $m\to\infty,$ the statement follows. ∎

Remark 3.4*.*

Here and in the sequel, the assumption $k\epsilon_{k}\geq 1,k\in\mathbb{N},$ is of a purely technical nature and has been made to simplify our exposition.

Recall from Section 2 that the set of aperiodic polynomials is dense in any $\mathcal{A}^{+}(w)$. Thus Theorem 3.3 implies the following statement.

Corollary 3.5**.**

Let $(\epsilon_{k})_{k=1}^{\infty}\subset(0,\infty)$ satisfy $\underline{\lim}_{k\to\infty}\,\epsilon_{k}=0$ and $k\epsilon_{k}\geq 1,$ $k\in\mathbb{N},$ and let $\tilde{w}=(k\epsilon_{k})_{k=1}^{\infty}.$ Then for every $f\in\mathcal{A}({\tilde{w}})$ there exists a sequence of polynomials $(Q_{m})_{m=1}^{\infty}\in\mathcal{A}^{+}$ such that

[TABLE]

The next result is our basic statement, allowing one to spread out the upper estimates for $|1-Q_{m}|$ proved in Theorem 3.3 from the sequence $(\theta_{m})_{m=1}^{\infty}$ to a larger set containing it. The result will help us to provide counterexamples on $L^{p}$-integrability of $(1-z)(1-F)^{-1}.$

Theorem 3.6**.**

Let $\psi:(0,1]\mapsto(0,\infty)$ and $\chi:(0,1]\mapsto(0,\infty)$ be continuous functions satisfying

[TABLE]

Then for every polynomial $P\in\mathcal{A}^{+}$ there exist a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ decreasing to zero and a sequence of polynomials $(Q_{m})_{m=1}^{\infty}\subset\mathcal{A}^{+}$ such that for all $m\in\mathbb{N}:$

• (i)

$m\theta_{m}\leq 2\pi,$**

• (ii)

$|1-Q_{m}(\theta_{m})|\leq 2\psi(\theta_{m})\theta_{m},$**

• (iii)

$\|P-Q_{m}\|_{\mathcal{A}}\leq\frac{2\theta_{m}}{\chi(\theta_{m})}.$**

Moreover, for each $m\in\mathbb{N}$ and for each $\theta$ such that $|\theta-\theta_{m}|\leq\psi(\theta_{m})\chi(\theta_{m})\theta_{m}$ one has

[TABLE]

Proof.

Let a polynomial $P$ be fixed, and let ${\rm deg}\,P=n.$ From (3.6) it follows that $\lim_{\theta\to 0+}\chi(\theta)=0,$ and then $\lim_{\theta\to 0+}\chi(\theta)\psi(\theta)=0.$ Since we have $\lim_{\theta\to 0+}\chi(\theta)(\psi(\theta))^{-1}=0$ as well, there exists $\Theta_{0}\in(0,1]$ such that

[TABLE]

Define

[TABLE]

and note that, in particular, $n\theta\leq 1,$ and $\gamma(\theta)\leq 1$ for all $\theta\in(0,\Theta_{0}].$

Moreover, if $\theta\in(0,\Theta_{0}]$, then by (3.7),

[TABLE]

Since $\gamma$ is continuous on $(0,\Theta_{0}]$ and $\lim_{\theta\to 0+}\,\tau(\theta)=+\infty,$ there exists a sequence $(\theta_{m})_{m\geq m_{0}}\subset[0,\Theta_{0}]$ satisfying

[TABLE]

Moreover, as $\lim_{\theta\to 0+}\gamma(\theta)=0$, we may also assume that $m\theta_{m}\leq 2\pi.$

Next, we fix $m\geq\max(m_{0},n),$ set $\theta_{0}=\theta_{m}$ and $\gamma=\gamma(\theta_{m}),$ and apply Corollary 3.2 to the polynomial $P.$ By (3.8), we infer that there exist $d_{m}>0$ and a polynomial $Q_{m}=P_{d_{m},\gamma(\theta_{m})}\in\mathcal{A}^{+}$ such that

[TABLE]

and

[TABLE]

Moreover, by (3.3) and (3.8),

[TABLE]

Using

[TABLE]

and (3.9), we conclude that if $|\theta-\theta_{m}|\leq\psi(\theta_{m})\chi(\theta_{m})\theta_{m},$ then

[TABLE]

∎

Remark 3.7*.*

Let $P\in\mathcal{A}^{+}$ and let $Q_{m}\in\mathcal{A}^{+}$ be polynomials given by Theorem 3.6. If $(w_{k})_{k=1}^{\infty}\subset[1,\infty)$ is a nondecreasing sequence, then the estimate from Theorem 3.6, (iii) yields

[TABLE]

In particular, if $w_{k}=k^{\nu}$, $k\in\mathbb{N},$ for some $\nu\in(0,1)$, then in view of $\theta_{m}m\leq 2\pi$, $m\in\mathbb{N}$, one has

[TABLE]

It will be convenient to separate the next easy corollary of Theorem 3.6

Corollary 3.8**.**

Let $\nu\in[0,1)$ and let $\varphi:(0,1]\mapsto(0,\infty)$ be a continuous function such that

[TABLE]

Then for every polynomial $P\in\mathcal{A}^{+}$ there exist a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ decreasing to zero and a sequence of polynomials $(Q_{m})_{m=1}^{\infty}\subset\mathcal{A}^{+}$ such that for all $m\in\mathbb{N}:$

• (i)

$\theta_{m}m\leq 2\pi,$**

• (ii)

$|1-Q_{m}(\theta_{m})|\leq 2\varphi(\theta_{m})\theta_{m}^{2-\nu},$**

• (iii)

$\|P-Q_{m}\|_{\mathcal{A}(\nu)}\leq\frac{2(2\pi)^{\nu}}{\varphi^{1/2}(\theta_{m})}.$**

Moreover, for each $m\in\mathbb{N}$ and each $\theta$ such that $|\theta-\theta_{m}|\leq\varphi^{3/2}(\theta_{m})\theta_{m}^{3-2\nu}$ one has

[TABLE]

Proof.

Define

[TABLE]

Since $\chi$ and $\psi$ satisfy (3.6), the corollary follows from Theorem 3.6 and Remark 3.7. ∎

By density arguments, the next result follows directly from Corollary 3.8.

Corollary 3.9**.**

Let $\varphi:(0,1]\mapsto(0,\infty)$ be a continuous function satisfying (3.11), and let $\nu\in[0,1).$ Then for every $f\in\mathcal{A}^{+}(\nu)$ there exist a sequence of polynomials $(Q_{m})_{m=1}^{\infty}\subset\mathcal{A}^{+}(\nu)$, and a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ decreasing to zero, such that

[TABLE]

and

[TABLE]

4. Main results

Using our construction of exponential polynomials from the previous section, we now produce a dense set of functions $F$ “almost” satisfying the Erdős-Feller-Pollard’s condition (1.3) but having such a strong singularity at $1$ that $(1-z)(1-F)^{-1}$ is unbounded in $\mathbb{D}.$ To this aim, we employ either Baire-category arguments (as in Theorem 4.2) or, alternatively, an iterative procedure (as in Theorem 4.4). Thus, we show that the Wiener-type condition (1.3) is the best one can hope for as far as the boundedness of $(1-z)(1-F)^{-1}$ is concerned. We then use power weights $w=(k^{\nu})_{k=1}^{\infty},\nu\in[0,1),$ to construct examples of $f\in\mathcal{A}^{+}(\nu)$ close to the constant function $1$ on a sufficiently large set (but violating (1.3)). This will be used in the next section to study the property $(1-z)(1-F)^{-1}\in L^{p}(\mathbb{T})$ for a fixed $p\in(1,\infty)$ in terms of the Taylor coefficients of $F.$

Proposition 4.1**.**

Let $L_{\theta}:X\mapsto(0,\infty),\theta\in(0,1],$ be a family of continuous functionals on a complete metric space $(X,\rho),$ and let $c\geq 0.$ Suppose that for any $f\in X$ there exists a sequence $(f_{m})_{m=1}^{\infty}\subset X$ satisfying

[TABLE]

Then there exists a residual set $S\subset X$ (in particular, dense in $X$) such that

[TABLE]

Proof.

Define a functional $\mathcal{F}_{m}$, $m\in\mathbb{N}$, on $X$ by

[TABLE]

Note that

[TABLE]

Since $(L_{\theta})_{\theta\in(0,1]}$ are continuous, $\mathcal{F}_{m}$ is upper-semicontinuous on $X$ for each $m\in\mathbb{N},$ by a standard argument. By e.g. [15, Theorem 9.17.3] for every $m\in\mathbb{N}$ the set $S_{m}$ of continuity points of $\mathcal{F}_{m}$ is residual. Hence

[TABLE]

is residual as well. Thus, by (4.1) and (4.3), for every $f\in S$ and every $m\in\mathbb{N}:$

[TABLE]

and the statement follows. ∎

Theorem 4.2**.**

Let $\nu\in[0,1)$, and let $\varphi:(0,1]\mapsto(0,\infty)$ be a continuous function satisfying (3.11). Then there exists a residual set $S\subset\mathcal{A}^{+}(\nu)$ of aperiodic functions with the following property: for every $f\in S$ there is a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ decreasing to zero such that

[TABLE]

Proof.

Consider a family $(L_{\theta_{0}})_{\theta_{0}\in(0,1]}$ of continuous functionals on $\mathcal{A}^{+}(\nu)$ given by

[TABLE]

for each $\theta_{0}\in(0,1].$ Using (2.1) and Corollary 3.9, we infer that $(L_{\theta})_{\theta\in(0,1]}$ satisfies the assumptions of Proposition 4.1 with $c=10.$ Therefore, there exists a residual set in $\mathcal{A}^{+}(\nu)$ satisfying (4.2) with $c=10.$ Note that the set of aperiodic functions in $\mathcal{A}^{+}(\nu)$ is residual. Taking the intersection of the two sets, we obtain a residual set satisfying (4.2) with $c=10$ again.

∎

Similarly, Corollary 3.5 and Proposition 4.1 imply the following statement.

Theorem 4.3**.**

Let $(\epsilon_{k})_{k=1}^{\infty}$ be a positive sequence such that

[TABLE]

If $\tilde{w}=(k\epsilon_{k})_{k=1}^{\infty}$ and $f\in\mathcal{A}^{+}(\tilde{w}),$ then for each $\epsilon>0$ there exists an aperiodic function $g\in\mathcal{A}^{+}({\tilde{w}})$ such that

[TABLE]

for some sequence $(\theta_{m})_{m=1}^{\infty}$ decreasing to zero.

Now we present another approach to Theorem 4.2 avoiding category arguments. Although the approach leads to a slightly weaker statement, it seems more transparent.

Theorem 4.4**.**

Let $\nu\in[0,1)$ be fixed, and let $\varphi$ satisfy (3.11). If $f\in\mathcal{A}^{+}(\nu),$ then for all $\epsilon>0$ and $\delta>0$ there exist an aperiodic function $g\in\mathcal{A}^{+}(\nu)$ and a sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ decreasing to zero such that $\|f-g\|_{{\mathcal{A}}(\nu)}\leq\epsilon$ and

[TABLE]

Proof.

Let $\epsilon>0$ and $\delta>0$ be fixed. Without loss of generality we may assume that $f=P_{0}$ is an aperiodic polynomial and that $\epsilon\in(0,1)$ is so small that the ball

[TABLE]

consists of aperiodic functions.

We construct the sequences

[TABLE]

with

[TABLE]

and the sequence of polynomials $(P_{m})_{m=1}^{\infty}\subset\mathcal{A}^{+}(\nu)$ so that, for $1\leq m\leq N$ and $\theta$ satisfying $|\theta-\theta_{m}|\leq\varphi^{3/2}(\theta_{m})\theta_{m}^{3-2\nu},$ one has

[TABLE]

and moreover

[TABLE]

Let

[TABLE]

By Corollary 3.8, there exists (large enough) $\theta_{1}\in(0,1)$ and a polynomial $P_{1}\in\mathcal{A}^{+}(\nu)$ such that

[TABLE]

whenever $|\theta-\theta_{1}|\leq\varphi^{3/2}(\theta_{1})\theta_{1}^{2-2\nu},$ and

[TABLE]

So, (4.9) and (4.10) hold for $N=1$.

Arguing by induction, suppose that (4.9) and (4.10) are true for some $N\geq 1$. Choose $\delta_{N+1}$ satisfying

[TABLE]

Then by Corollary 3.8 applied to $P=P_{N}$ there exist $P_{N+1}\in\mathcal{A}^{+}(\nu)$, and (small enough) $\theta_{N+1}<\theta_{N}$ such that

[TABLE]

if $|\theta-\theta_{N+1}|\leq\varphi^{3/2}(\theta_{N+1})\theta_{N+1}^{3-2\nu},$ and, moreover,

[TABLE]

Hence for every $m\in[1,N]$ and every $\theta$ satisfying $|\theta-\theta_{m}|\leq\varphi^{3/2}(\theta_{m})\theta_{m}^{3-2\nu}$ we have

[TABLE]

Taking in account (4.11), we infer that (4.9) and (4.10) hold for $N+1$ too.

By (4.8) the sequence $(P_{m})_{m=0}^{\infty}$ is Cauchy in $\mathcal{A}^{+}(\nu),$ so there exists $g\in\mathcal{A}^{+}(\nu)$ such that

[TABLE]

Finally, (4.9) yields (4.7), and moreover $g$ is aperiodic by the above. ∎

5. Regularity of reciprocals in terms of the $L^{p}$-scale

In this section we will study the regularity of generating functions for renewal sequences with respect to the $L^{p}$-scale. Namely, we will be concerned with the identifying $p\geq 1$ such that $(1-z)(1-F)^{-1}\in L^{p}(\mathbb{T}),$ where $F(z)=\sum_{k=1}^{\infty}a_{k}z^{k},z\in\mathbb{D},$ is aperiodic. It is clear that if

[TABLE]

where $f(\theta)=F(e^{i\theta}),$ then $(1-z)(1-F)^{-1}\in L^{p}(\mathbb{T})$ if and only if $R_{f}\in L^{p}[-\pi,\pi],p\geq 1.$ Thus, it is enough to study the same issue for the function $R_{f},$ and the results on $(1-z)(1-F)^{-1}$ below will be formulated in terms of $R_{f}.$ We will show, in particular, that while $R_{f}$ has a certain amount of regularity, being in $L^{p}[-\pi,\pi]$ for $p\in[1,2],$ $R_{f}$ is not very regular in the sense that $R_{f}$ does not belong, in general, to $L^{p}[-\pi,\pi],$ if $p>3.$ This result will be put below into a more general (and sharper) context of the spaces $\mathcal{A}^{+}(\nu).$

To get positive results on the regularity of $R_{f}$ we will use an idea from [27]. In particular, we will use the following crucial result proved in [27, Theorem 1]. (The result formulated in [27] has a weaker form but the proof given there yields the statement given below.)

Theorem 5.1**.**

Let

[TABLE]

and

[TABLE]

with $\mu_{k}\geq 1$ and $\alpha_{k}\in\mathbb{R},k\geq 1.$ Then

[TABLE]

Recall that here and in the sequel meas stands for the Lebesgue measure.

Remark 5.2*.*

Note that the above estimate is the best possible as the example of $f(\theta)=\cos\theta$ shows. Remark also that Theorem 5.1 was stated in [27] with a constant $A$ instead of $4\pi$ above. The uniformity of $A$ was not clarified in [27]. Since that property is crucial for our reasoning and to be on a safe side we provide an independent proof of Theorem 5.1 in Appendix A.

Corollary 5.3**.**

Let $(a_{k})_{k\geq m}\subset[0,\infty),m\in\mathbb{N},$ and let $r=\sum_{k=m}^{\infty}a_{k}\in(0,1].$ If

[TABLE]

then for every $\theta_{0}\in(0,1]$ such that $m\theta_{0}\leq 1,$ one has

[TABLE]

Proof.

If $\tilde{f}(t)=r^{-1}f(\theta_{0}(t+1))$ and $\epsilon\in(0,r]$ then by (5.1) we obtain:

[TABLE]

∎

We will also need the next technical estimate for distribution functions. Its proof is postponed to Appendix A.

Proposition 5.4**.**

Let $\Omega\subset[0,\infty)$ be a measurable set of finite measure, and let $\varphi:\Omega\mapsto(0,\infty)$ be a measurable function. Suppose that there are constants $r>\eta>0$ and $A>0$ such that $\eta\leq\varphi\leq r$ a.e. on $\Omega$ and, moreover,

[TABLE]

Then for all $d\geq 0$ and $p\geq 1,$

[TABLE]

Now we are ready to prove one of the main results of this section. It is an extension and sharpening of Littlewood’s Theorem $2$ from [27]. For

[TABLE]

from $\mathcal{A}^{+},$ define for $\theta\in(0,1]:$

[TABLE]

Note that

[TABLE]

Theorem 5.5**.**

Let $f\in\mathcal{A}^{+}$ be given by (5.4) and let $\theta_{0}\in(0,1]$. Then for each $p\geq 1$ there exists $C_{p}>0$ such that for every $\theta_{0}\in(0,1]$ with $r(\theta_{0})<1$ one has:

[TABLE]

Proof.

Let $n=n(\theta_{0})$ satisfy $n\leq 1/\theta_{0}<n+1,$ and let $p\geq 1$ be fixed. By assumption,

[TABLE]

If $\theta\in[\theta_{0},2\theta_{0}]$ and $1\leq k\leq n$ then $0<k\theta\leq 2n\theta_{0}<3\pi/4$ so that $\sin k\theta\geq k\theta/4$ and $\sin k\theta/2\geq k\theta/\pi$. Hence for $\theta\in[\theta_{0},2\theta_{0}]$ we have

[TABLE]

and

[TABLE]

If $r(\theta_{0})=0$, then by (5.8) we have

[TABLE]

and (5.6) holds.

Let now $r(\theta_{0})>0$. For $y\in[0,r(\theta_{0})]$ let $S(y)$ be the set of such $\theta\in[\theta_{0},2\theta_{0}]$ that

[TABLE]

Corollary 5.3 then yields

[TABLE]

Now, since

[TABLE]

by (5.10) again, we have

[TABLE]

So, putting

[TABLE]

we infer that for every $\theta\in S(\eta):$

[TABLE]

We now estimate the left-hand side of (5.6) as follows. Write

[TABLE]

where

[TABLE]

We deal with each of the terms $J_{p,1}$ and $J_{p,2}$ above separately. First, observe that by (5.12),

[TABLE]

Second, if in addition $\eta\leq r(\theta_{0}),$ then

[TABLE]

Setting to simplify the notation

[TABLE]

and using Proposition 5.4 and (5.11), we obtain that

[TABLE]

Taking in account (5.13) we infer that (5.6) holds with $C_{p}=3p\cdot 2^{8p}.$ ∎

Theorem 5.5 allows us to describe the integrability of $1/(1-f)$ in terms of the size of Fourier coefficients of $f\in\mathcal{A}^{+}$. We will need the next simple proposition on series with positive terms, proved in Appendix A.

Proposition 5.6**.**

Let $(q_{k})_{k\geq 2^{m-1}+1},m\in\mathbb{N},$ be a positive decreasing sequence, and let $\alpha\geq 0$. Then

[TABLE]

For $f\in\mathcal{A}$ define

[TABLE]

Corollary 5.7**.**

Let $f\in\mathcal{A}^{+}$ be given by (5.4). Then there exist $c>0$ and $m_{0}\in\mathbb{N}$ such that

[TABLE]

In particular, if $f$ is aperiodic and the right-hand side of (5.14) is finite, then $1/(1-f)\in L^{1}[-\pi,\pi]$.

Proof.

Choose $\Theta_{0}\in(0,1]$ with $r(\Theta_{0})<1$. By Theorem 5.5,

[TABLE]

for a constant $c>0.$ Fix $n_{0}\in\mathbb{N}$ such that

[TABLE]

and note that $W_{n}>0,n\geq m_{0},$ and moreover $(W_{n})_{n=m_{0}}^{\infty}$ monotonically increases. Using (5.15) and Proposition 5.6, we obtain:

[TABLE]

[TABLE]

that is, the right-hand side estimate in (5.14) holds. If the series $\sum_{n=m_{0}}^{\infty}{1}/(nW_{n})$ converges, then since $f$ is aperiodic and $|R_{f}(\theta)|$ is symmetric in the sense that $|R_{f}(\theta)|=|R_{f}(-\theta)|,$ we infer that $1/(1-f)\in L^{1}[-\pi,\pi].$

To prove the left-hand side estimate in (5.14), we note that if $n\in\mathbb{N}$ and $\theta\in(0,1]$ are such that $n\leq 1/\theta<n+1$, then, using (5.5), we have

[TABLE]

Therefore, from (5.16) it follows that there exists $m_{0}\in\mathbb{N}$ such that

[TABLE]

(Since we do not need Theorem 5.5 for the left-hand side estimate, a dyadic partition of $[0,1/m_{0}]$ is replaced with a partition, in a sense, more convenient for writing down the final estimate.) ∎

Remark 5.8*.*

From (5.15) it follows that if $f\in\mathcal{A}^{+}$ is given by (5.4) and

[TABLE]

then

[TABLE]

cf. Littlewood’s result (1.7).

If $\mu<\infty,$ then

[TABLE]

for every $\delta>0,$ since in this case

[TABLE]

and

[TABLE]

On the other hand, in this case (5.17) is a direct consequence of $\lim_{\theta\to 0+}\,(f(\theta)-1)/\theta=\mu.$

Remark 5.9*.*

For aperiodic $f(\theta)=\sum_{k=1}^{\infty}a_{k}e^{ik\theta}\in\mathcal{A}^{+}$, let $F(z):=\sum_{k=1}^{\infty}a_{k}z^{k},z\in\overline{\mathbb{D}}.$ Recall that $1/(1-F)$ is analytic in $\mathbb{D}$ and continuous in ${\mathbb{D}}\setminus\{1\}.$ If $1/(1-f)\in L^{1}[-\pi,\pi],$ then $1/(1-F)\in L^{1}(\mathbb{T}),$ hence $1/(1-F)$ belongs to the Hardy space $H^{1}(\mathbb{D}).$ If $1/(1-F(z))=\sum_{k=0}^{\infty}b_{k}z^{k},z\in\mathbb{D},$ then by Hardy’s inequality ([18, IX.9.7]),

[TABLE]

We pause now to illustrate Corollary 5.7 by the following example.

Example 5.10*.*

Consider

[TABLE]

Then $f_{\epsilon}\in\mathcal{A}^{+},$ and it is aperiodic. For each $\epsilon>0$ we have

[TABLE]

hence $1/(1-f_{\epsilon})\in L^{1}[-\pi,\pi]$ by (5.14). On the other hand, if $\epsilon=0$ then

[TABLE]

for some constant $c>0,$ so (5.14) implies that $1/(1-f_{0})\not\in L^{1}[-\pi,\pi]$. (Similarly, if $f\in\mathcal{A}^{+}$ is given by

[TABLE]

then $f$ is aperiodic and $1/(1-f)\not\in L^{1}[-\pi,\pi]$.)

Although, in general, $R_{f}\not\in L^{p}[-\pi,\pi],p>3,$ for $f\in{\mathcal{A}}^{+},$ it is possible to formulate a sufficient condition on the Fourier coefficients $f$ ensuring $R_{f}\in L^{p}[-\pi,\pi]$ for fixed $p>2.$ (This way we may also produce $f\in{\mathcal{A}}^{+}$ such that $R_{f}\in\cap_{p>2}L^{p}[-\pi,\pi].$) The next statement is a direct implication of Theorem 5.5.

Corollary 5.11**.**

Let $f\in\mathcal{A}^{+}$ be aperiodic, and let $p>2.$ If there is $m_{0}\in\mathbb{N}$ such that

[TABLE]

then $R_{f}\in L^{p}[-\pi,\pi]$.

Proof.

By aperiodicity $f$ and symmetry of $|R_{f}(\theta)|$ it suffices to prove that

[TABLE]

for some $\delta>0$. Using (5.18), Theorem 5.5 and Proposition 5.6, we infer that there exists $c>0$ such that for large enough $n_{0}$ and $m_{0}=2^{n_{0}-1}+1$:

[TABLE]

∎

Further we will make use of Theorem 5.5 to describe the regularity $R_{f}$ with respect to the $L^{p}$-scale for arbitrary $f\in\mathcal{A}^{+}(\nu),\nu>0.$

Corollary 5.12**.**

Let $\nu\in(0,1),$ and $f\in\mathcal{A}^{+}(\nu)$ be aperiodic. Then $R_{f}\in L^{p}[-\pi,\pi],$ where $p=1+\frac{1}{1-\nu}.$ On the other hand, for any $\nu\in[0,1)$ and

[TABLE]

there exists an aperiodic function $f\in\mathcal{A}^{+}(\nu)$ such that $R_{f}\not\in L^{p}[-\pi,\pi].$

Proof.

Let us prove the first claim. Suppose $f(\theta)=\sum_{k=1}^{\infty}a_{k}e^{ik\theta},\theta\in[-\pi,\pi].$ Again, by aperiodicity of $f$ and symmetry of $|R_{f}(\theta)|$ it suffices to show that that (5.19) holds for some $\delta>0$ and $p=1+1/(1-\nu).$ Choose $n_{0}\in\mathbb{N}$ so large that $W_{n}\geq 1/2$, $n\geq n_{0}.$ Then, taking into account [18, Thm. 165], we conclude that there are constants $c>0$ and $\tilde{c}>0$ such that

[TABLE]

Thus, Corollary 5.11 implies that $R_{f}\in L^{p}[-\pi,\pi].$

Second, to prove the negative result, let $p\in\left(2+\frac{1}{1-\nu},\infty\right)$ be fixed. Write

[TABLE]

and for a fixed $\epsilon\in(0,1-\nu)$ define a continuous function

[TABLE]

Since $\varphi$ satisfies (3.11), Theorem 4.2 implies that there exists an aperiodic $f=f_{\epsilon}\in\mathcal{A}^{+}(\nu)$ and a decreasing sequence $(\theta_{m})_{m=1}^{\infty}\subset(0,1]$ tending to zero such that

[TABLE]

whenever $|\theta-\theta_{m}|\leq\theta_{m}^{3-2\nu}.$ Setting

[TABLE]

we have

[TABLE]

for some constant $c>0.$ Note that

[TABLE]

and choose $\epsilon$ such that $\beta>0,$ that is

[TABLE]

As $\theta_{m}\to 0$, $m\to\infty$, the right-hand side of (5.20) tends to infinity as $m\to\infty,$ hence $R_{f}\not\in L^{p}[-\pi,\pi].$ ∎

Finally, as a consequence of Corollary 5.12, we derive a result on the regularity of $R_{f}$ measured in terms of $L^{p}$-spaces. The result corresponds formally to the case $\nu=0$ in Corollary 5.12 and should be compared to the property (1.5) discussed by Erdős, de Bruijn and Kingman. While its positive part is elementary, it was apparently overlooked by specialists in probability theory.

Corollary 5.13**.**

Let $f(\theta)=\sum_{k=1}^{\infty}a_{k}e^{ik\theta}\in\mathcal{A}^{+}$ be aperiodic, and let $(b_{k})_{k=0}^{\infty}$ be a renewal sequence associated to $(a_{k})_{k=1}^{\infty}.$ Then $R_{f}\in L^{2}[-\pi,\pi]$ and

[TABLE]

At the same time, there exists an aperiodic $f\in\mathcal{A}^{+}$ such that $R_{f}\not\in L^{p}[-\pi,\pi]$ for every $p\in(3,\infty].$

Proof.

The second claim follows directly from Corollary 5.12. Let $F(z):=\sum_{k=1}^{\infty}a_{k}z^{k},z\in\overline{\mathbb{D}},$ so that $f(\theta)=F(e^{i\theta}),\theta\in[-\pi,\pi].$ Then $1/(1-F(z))=\sum_{k=0}^{\infty}b_{k}z^{k},z\in\mathbb{D}.$ Note that ${\rm Re}\,(1/(1-F))\in L^{1}(\mathbb{T})$ since

[TABLE]

by Fatou’s Lemma and positivity of the harmonic function ${\rm Re}\,(1/(1-F))$ in $\mathbb{D}$, cf. [23, p. 10-12]. On the other hand, if $n_{0}$ is such that $a_{n_{0}}\not=0$ and $\theta\in[-1/n_{0},1/n_{0}],$ then

[TABLE]

and

[TABLE]

Thus $R_{f}\in L^{2}[-1/n_{0},1/n_{0}],$ and, if $f$ is aperiodic, then $R_{f}\in L^{2}[-\pi,\pi].$ As the latter property is equivalent to $(1-z)(1-F)^{-1}\in L^{2}(\mathbb{T})$, Parseval’s identity yields (5.21). ∎

Remark 5.14*.*

For $f\in\mathcal{A}^{+}$ and $G_{\epsilon}:=\{\theta\in[-\pi,\pi]:{\rm Re}\,f(\theta)\geq 1-\epsilon\}$ there are several estimates in the literature of the form

[TABLE]

with an absolute constant $C>0,$ see e.g. [4], [9], [10], [25], [30]. The estimates are motivated by applications in probability theory and number theory, and they seem to be weaker than the estimate (5.1) provided by Theorem 5.1. (A related bound for $f$ in terms of its coefficients has been given in [5].) To clarify their relations to our treatment, for $\epsilon\in(0,1]$ and $f\in\mathcal{A}^{+}$ define

[TABLE]

Note that

[TABLE]

On the other hand, if $f$ is aperiodic then Littlewood’s theorem (1.8) implies that for every $\alpha\in(0,1)$ there exists $c_{\alpha}(f)>0$ such that

[TABLE]

Indeed, let $\alpha\in(0,1)$ be fixed. Then, taking $\gamma\in(0,1-\alpha)$ and using (1.8), we have

[TABLE]

and (5.22) follows. Thus (1.8) gives asymptotically better bounds for ${\rm meas}\,(E_{\epsilon}).$ However, it is not clear whether the constant $c_{\alpha}(f)$ can be taken independent of $f.$

6. Remarks on $p$-functions

The theory of $p$-functions can be considered as a continuous counterpart of the theory of renewal sequences. Its basic facts can be found in [23]. To put the relevant considerations on $p$-functions into our setting, let us first recall a couple of basic facts.

Let a function $\varphi$ in the right half-plane $\mathbb{C}_{+}:=\{z\in\mathbb{C}:{\rm Re}\,z>0\}$ be defined as

[TABLE]

where $c\geq 0,$ and $\nu$ is a positive Borel measure on $(0,\infty)$ satisfying

[TABLE]

(It is easy to see that $\varphi$ is analytic in $\mathbb{C}_{+}$ and continuous in $\overline{\mathbb{C}}_{+}.$) Then there is a unique continuous function $g:[0,\infty)\mapsto[0,1],g(0)=1,$ called a standard $p$-function, such that the Laplace transform $\mathcal{L}g$ of $g$ can be represented as

[TABLE]

for $z>0.$ Observe that if $g$ is a standard $p$-function, then $\mathcal{L}g$ extends analytically to $\mathbb{C}_{+}$ and continuously to $\overline{\mathbb{C}}_{+}\setminus\{0\},$ see [23, p. 74] and [21, Theorem 5]. For $p$-function $g$ given by (6.2) and (6.1) we will write $g\sim(c,\nu)$.

We refer to [23, Chapter 3] concerning basic facts of the analytic theory of $p$-functions. Note that $\varphi$ in (6.1) is a so-called Bernstein function, and the class of $p$-functions is an important subclass of a class of potential measures arising in the study of Bernstein functions. For a thorough discussion of Bernstein functions and associated potential measures, see [31, Ch. 5, p. 63-64 and Ch. 11].

Moreover, by [21, Theorem 6], for $g\sim(c,\nu)$ one has

[TABLE]

It was proved in [22, Theorem 3] (see also [23, p. 75-76]) that if $g(\infty)>0$, i.e. if

[TABLE]

then $g$ has bounded variation on $[0,\infty).$ As in the setting of renewal sequences, a natural question is whether $g$ has always bounded variation, i.e. also in the case when $g(\infty)=0.$ The question was asked by J. Kingman in [23, p. 76], and soon after J. Hawkes produced in [17] an example showing that the answer is “no” in general.

The argument in [17] was based on the following observation. Let

[TABLE]

and the corresponding $p$-function $g$ be given by

[TABLE]

If $g$ has bounded variation on $[0,\infty)$ and $g(\infty)=0,$ then

[TABLE]

Essentially, Hawkes constructed a quasi-exponential series

[TABLE]

(see the introduction) such that

[TABLE]

(thus ${\lim}_{\theta\to 0+}\,{|1-f(\theta)|}/{\theta}$ does not exist ). Then, setting

[TABLE]

for $z$ with ${\rm Re}\,z\geq 0,$ one obtains the desired (counter-)example.

One can prove that in Hawkes’s example

[TABLE]

In other words, the example states that there exists a finite (discrete) Borel measure $\nu$ on $(0,\infty)$, satisfying

[TABLE]

such that the corresponding $p$-function $g\sim(0,\nu)$ has unbounded variation on $[0,\infty).$ If one can arrange ${\rm supp}\,\nu\subset\mathbb{N},$ then the example could also be used to produce a negative answer to the question by Erdős-de Bruijn-Kingman on renewal sequences. However, we do not see how to realize that in a way different from the above.

On the other hand, using our results, we can generalize the considerations by Hawkes in the following way, thus showing that the condition (6.3) is best possible in a sense (as in (1.3), the discrete analogue of (6.3)).

Theorem 6.1**.**

Let $\beta:(0,\infty)\to(0,\infty)$ be a function satisfying

[TABLE]

Then there exists a finite (discrete) Borel measure $\nu=\nu_{\beta}$ on $(0,\infty)$ such that

[TABLE]

and the corresponding $p$-function $g\sim(0,\nu)$ has unbounded variation on $[0,\infty)$.

Proof.

By Theorem 4.3, setting $w=(k\beta(k))_{k=1}^{\infty},$ there exists $f\in\mathcal{A}^{+}(w)$ such that

[TABLE]

If $F(e^{i\theta})=f(\theta),$ then $z\to F(e^{-z})$ is of the form (6.1) for an appropriate discrete measure $\nu$ supported by $\mathbb{N}.$ If a $p$-function $g$ is defined by

[TABLE]

then $g$ satisfies (6.6), and, moreover, it has unbounded variation on $[0,\infty)$ by (6.4) and (6.7).

∎

7. Appendix A: Technicalities

The proof of Lemma 3.1   Since $\lambda/2\in(0,\pi/2)$, $\gamma/2\in(0,\pi/2)$ and $(\gamma+\lambda)/2\in(0,\pi/3)$, we have

[TABLE]

Moreover, as

[TABLE]

and

[TABLE]

for $a,b\in\mathbb{R},$ we obtain

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

and then

[TABLE]

$\Box$

To prove Littlewood’s result mentioned in Section 5 with an explicit constant, we prove first the next auxiliary estimate; see also Remark 5.2.

Lemma 7.1**.**

If $E\subset[a,a+2\pi]$ is a measurable set such that

[TABLE]

for some $\epsilon>0,$ then

[TABLE]

Proof.

By $2\pi$-periodicity of $\cos t,$ we may assume without loss of generality that $E\subset[-\pi,\pi].$ Write $I=[-{\rm meas}(E)/2,{\rm meas}\,(E)/2]$ and

[TABLE]

Noting that ${\rm meas}\,(\tilde{E})={\rm meas}\,(E_{2}),$ we have

[TABLE]

hence

[TABLE]

so that, as

[TABLE]

we obtain

[TABLE]

and the statement follows. ∎

Corollary 7.2**.**

Let $\alpha\geq 1$, $\beta\in\mathbb{R},\epsilon>0,$ and let a measurable set $E\subset[a,a+2\pi],a\in\mathbb{R},$ be such that

[TABLE]

Then

[TABLE]

Proof.

We have

[TABLE]

where $G=\alpha E+\beta\subset[\alpha a+\beta,\alpha a+\beta+2\pi\alpha]$ and ${\rm meas}\,(G)=\alpha\,{\rm meas}\,(E)$ $\leq 2\pi\alpha.$ Set $n=[\alpha+1],$ $h=2\pi\alpha/n,$ and write

[TABLE]

so that

[TABLE]

By assumption,

[TABLE]

Hence, by Lemma 7.1 and convexity arguments,

[TABLE]

∎

Thus, we have the following result, formulated essentially in [27, Lemma] with a constant $A$ instead of $(16\pi^{2})^{1/3}$.

Lemma 7.3**.**

Let

[TABLE]

where $\alpha_{m}\geq 1$, $\beta_{m}\in\mathbb{R},$ $m\in\mathbb{N}$, and

[TABLE]

If $\epsilon>0$ and $E\subset[-\pi,\pi]$ is a measurable set such that

[TABLE]

then ${\rm meas}\,(E)\leq(16\pi^{2})^{1/3}\epsilon^{1/3}$.

Now we are able to prove Theorem 5.1 with the absolute constant $4\pi.$

Proof of Theorem 5.1.   Let $f$ be as in Theorem 5.1, and let $E_{\epsilon}:=\{t\in[-\pi,\pi]:f(t)\geq 1-\epsilon\}.$ Then

[TABLE]

and from Lemma 7.3 it follows that ${\rm meas}\,(E_{\epsilon})\leq(16\pi^{2})^{1/3}\epsilon^{1/3}({\rm meas}\,(E_{\epsilon}))^{1/3},$ hence

[TABLE]

$\Box$

We now turn to the proof of Proposition 5.4. First, let us recall one of the forms of layer-cake representation, which is a direct consequence of e.g. [14, Prop. 1.3.4] or [26, Ch. 1.13].

Lemma 7.4**.**

Let $\psi:[0,\infty)\mapsto(0,\infty)$ be a differentiable strictly decreasing function with $\lim_{t\to\infty}\,\psi(t)=0$, and let $\varphi:\Omega\mapsto(0,\infty)$ be a measurable function, where $\Omega\subset[0,\infty)$ is a measurable set of finite measure. Then

[TABLE]

where

[TABLE]

(It suffices to note that

[TABLE]

and apply either of the statements from [14] or [26] to $\psi(0)-\psi(t)$.)

Having Lemma 7.4 in mind, we are now ready to give a proof of Proposition 5.4.

Proof of Proposition 5.4.    Let $g_{\varphi}$ be defined by (7.4) so that

[TABLE]

By assumption,

[TABLE]

Using Corollary 7.4 with $\psi(t)=1/(t+d)^{p},$ we obtain

[TABLE]

Next, using the estimate (see e.g. [18, Thm. 41])

[TABLE]

and the elementary inequality

[TABLE]

we have

[TABLE]

From this and (7.5) we obtain (5.3). $\Box$

We finish this section with the proof of auxiliary Proposition 5.6 on positive series.

Proof of Proposition 5.6.   Note that

[TABLE]

so

[TABLE]

and then

[TABLE]

On the other hand,

[TABLE]

hence

[TABLE]

and

[TABLE]

$\Box$

8. Appendix B: Croft’s approach

In this section we present a different proof of Corollary 3.8 based on Croft’s approach from [8] dealing with quasi-exponential series, that is with trigonometrical series with real frequencies. However, as we remarked above, the argument in [8] seems to be incomplete.

Let us briefly compare Croft’s approach with the one of the present paper. For a sufficiently small parameter $\theta_{0}$ both approaches aim at finding $d$ and $\gamma$ such that $\delta(\theta_{0})=|1-P_{d,\gamma}(\theta_{0})|$ is “small”, in particular, $\delta(\theta_{0})\leq c\theta_{0}^{2-\epsilon}$ for some $\epsilon\in(0,1],$ where a constant $c$ does not depend on $\theta_{0}.$ Croft proceeds by requiring ${\rm Im}\,P_{d,\gamma}(\theta_{0})=0.$ This way he expresses $\gamma$ in terms of $d$ and then chooses $d$ to ensure the inequality above. Unfortunately, his proof stops at this step. Proceeding in a different way, for a fixed $\gamma$ we minimize the quadratic function $q(d)=(1+d)^{2}|1-P_{d,\gamma}(\theta_{0})|^{2}$ with respect to $d.$ This relates $d$ to $\gamma,$ and allows us to make the quantity $\delta$ “small” enough to fit the steps of our inductive constructions in Section 3. The two steps lead eventually to similar estimates of $\delta(\theta_{0}).$ On the other hand, we also have to take care of a) getting polynomials with integer frequencies eventually, b) spreading out our estimates for $\delta$ to large sets, c) extending our estimates for polynomials $(Q_{m})_{m=1}^{\infty}$ from fixed $\theta_{0}$ to appropriate sequences of $(\theta_{m})_{m=1}^{\infty}\subset[0,\pi]$ going to [math] and then, finally, d) of constructing $f\in\mathcal{A}^{+}(w)$ out of $(Q_{m})_{m=1}^{\infty}$ via a limiting procedure.

Lemma 8.1**.**

Let

[TABLE]

$\epsilon\in(0,1],$* and $\alpha\in(0,\epsilon/2].$ Then for any $\theta_{0}\in(0,1/n^{1/\alpha}],$*

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

Proof.

Since $n\theta_{0}\in(0,1]$ and $n\leq\theta_{0}^{-\alpha}$,

[TABLE]

and, in view of $\epsilon\geq 2\alpha$,

[TABLE]

so that (8.1) and (8.2) hold. Moreover,

[TABLE]

and we get (8.3). ∎

As an illustration, to show that Croft’s idea actually works, we provide now a proof of Corollary 3.8 following Croft’s approach.

Proof of Corollary 3.8.    For $d>0$ and $\tau>0$, define

[TABLE]

Then

[TABLE]

and

[TABLE]

Fix $\Theta_{0}\in(0,1)$ such that

[TABLE]

so that, in particular,

[TABLE]

For $\theta\in(0,\Theta_{0}],$ define

[TABLE]

and note that

[TABLE]

Then the assumption $\theta\in(0,1/n^{1/\alpha(\theta)}]$ takes the form $n\leq\varphi^{1/4}(\theta)$ and holds for $\theta\in(0,\Theta_{0}]$, by (8.5).

Furthermore, for $\theta\in(0,\Theta_{0}]$ define

[TABLE]

By (8.1) and (8.6) we infer that

[TABLE]

hence

[TABLE]

is well-defined. Observe that $\tau$ is continuous on $(0,\Theta_{0}],$ and it satisfies $\lim_{\theta\to 0+}\,\tau(\theta)=+\infty$. Therefore, there exists a sequence $(\theta_{m})_{m\geq m_{0}}\subset(0,\Theta_{0}]$ such that

[TABLE]

We may assume that $\theta_{m+1}<\theta_{m}$. From (8.9) and (8.8) it follows that $m\theta_{m}\leq 2\pi,m\geq m_{0}.$ Moreover, the latter condition implies that $m\theta_{m}\in(3\pi/2,2\pi]$ for large $m$, so without loss of generality, we may assume that $m\theta_{m}\in(3\pi/2,2\pi],m\geq m_{0}.$

Define a polynomial $Q_{m}$ by (8.4) with $d=d(\theta_{m})$ and $\tau=\tau_{m}=m.$ Then, employing (8.8) and (8.9), we obtain that

[TABLE]

Moreover, by (8.2),

[TABLE]

and, in view of $m\theta_{m}\in(3\pi/2,2\pi],m\geq m_{0},$ and (8.7),

[TABLE]

Thus, taking into account (8.11) and (8.10),

[TABLE]

Next, to estimate the distance between $P$ and $Q_{m}$ we note that

[TABLE]

and then, by Remark 3.7,

[TABLE]

Finally, (8.3), (8.5), and (8.6) imply that

[TABLE]

Therefore, if $|\theta_{m}-\theta|\leq\varphi^{3/2}(\theta_{m})\theta_{m}^{3-2\nu},$ then

[TABLE]

$\Box$

9. Acknowledgement

The authors would like to thank J. Aaronson, who attracted the authors’ attention to the problem posed by Erdős, de Bruijn, and Kingman.