Generic power series on subsets of the unit disk
aa r X i v : . [ m a t h . C V ] J a n GENERIC POWER SERIES ON SUBSETS OF THE UNIT DISK
BAL ´AZS MAGA AND P ´ETER MAGAA
BSTRACT . In this paper, we examine the boundary behaviour of the generic power series f with coefficients chosen from afixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open set U with a non-real boundary pointon the unit circle, f p U q is a dense set of C . As it is demonstrated, this conclusion does not necessarily hold for arbitrary opensets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property isgiven.
1. I
NTRODUCTION
Let Λ Ď C be a bounded subset with at least two elements, endowed with its usual subspace topology. Moreover,assume that the product space Ω : “ Ω Λ : “ ą n “ Λ is a Baire space. Due to Alexandrov’s theorem (e.g. [4, p. 408]), it holds for example if Λ is G δ . These generalconditions on Λ will be assumed throughout the paper.For any p λ n q n “ P Ω we can define the power series f p z q : “ f p λ n q p z q : “ ÿ n “ λ n z n . The resulting function is clearly holomorphic in the open unit disk D : “ t| z | ă u . Roughly speaking, we are interestedin the generic behaviour of f in terms of Baire category near the boundary B D “ S .This work is a direct continuation of our previous paper [5], in which we investigated the typical boundary behaviourof real power series with coefficients chosen from a finite set Λ . (Actually, the set of coefficients was denoted by D inthat paper, we opted to introduce this notational modification as the symbol D is customarily preserved for the unit diskin this setup.) While the probabilistic aspects of the problem was our main focus (considering the uniform distributionover Λ ), we proved straightforward results in terms of Baire category as well ([5, Theorem 3]). Notably, if Λ hasboth positive and negative elements, for the generic power series f we have lim sup ´ f “ `8 and lim inf ´ f “ ´8 ,while if Λ Ď r , `8q (resp. Λ Ď p´8 , s ), we have lim ´ f “ `8 (resp. lim ´ f “ ´8 ). It was natural to considerthe same problem in the complex setup as well, which is a more natural habitat of power series. (We note that whilethose results were stated for finite Λ exclusively, the proofs can be generalized in a straightforward manner for any Λ for which Ś n “ Λ is a Baire space.)Another predecessor of this line of research is [1], in which results are stated and proved about generic complexpower series. For a more detailed historical summary of the topic we also refer to that paper: while the problem ofthe “general behaviour” dates back to Borel, the more thorougly examined probabilistic question was worked out byseveral authors, as presented in [2]. In terms of Baire category, the first results were provided in [3]. The setup of [3]slightly differs from ours, as the topological vector space H p D q in which Baire category is investigated is the spaceof all functions which are holomorphic in D with the topology of locally uniform convergence. (The Ω we consider Date : October 2020.2010
Mathematics Subject Classification.
Primary 30B30; Secondary 28A05, 54H05.
Key words and phrases. complex power series, boundary behaviour, Baire category.The first author was supported by the ´UNKP-20-3 New National Excellence Program of the Ministry for Innovation and Technology fromthe source of the National Research, Development and Innovation Fund, and by the Hungarian National Research, Development and InnovationOffice–NKFIH, Grant 124003. The second author was supported by the Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences,the MTA R´enyi Int´ezet Automorphic Research Group, and by the Hungarian National Research, Development and Innovation Office–NKFIH,Grants FK 135218, K 119528. orresponds to a subspace of it.) Their main results stated that generically, S is a natural boundary and f p D q “ C .These results were generalized by [1], in particular: Proposition 1 ([1, Proposition 3.2]) . For any U open set for which U X S ‰ H , we have f p U q “ C . (For a set H Ď C ,its closure is denoted by H throughout the paper.) The proof of this statement relies on Runge’s theorem on polynomial approximation, which is out of reach in oursetup which poses restrictions on the permissible holomorphic functions. Consequently, when one would like to verifysimilar results for Ω , different techniques are required. As we will see, this leads to somewhat weaker theorems:roughly, we could only verify that f p U q is an open, dense set instead of being equal to C . Our first main result is thefollowing. Theorem 1.
Assume that U Ď D is open with an accumulation point ζ P S with non-vanishing imaginary part. Thenfor the generic p λ n q n “ P Ω , the image f p U q Ď C is dense and open. As we will see below, the conclusion of Theorem 1 does not hold for all open sets accumulating to S . In orderto allow 1 or ´ Λ should be introduced. Thenecessary and sufficient conditions are summarised by our other theorems. By a real line, we mean a one-dimensionalreal affine subspace of C , and by a real half-plane we mean one of the two components of the complement of a realline. Theorem 2.
Assume that U Ď D is open and ´ P U. If Λ is not contained by a real line, the image f p U q Ď C is denseand open generically. Otherwise there exists such U for which f p U q evades a real half-plane, regardless of the choiceof p λ n q n “ . Theorem 3.
Assume that U Ď D is open and P U. If Λ is not contained by a half-plane of the form t z : α ď arg z ď α ` π u , then the image f p U q Ď C is dense and open generically. Otherwise there exists such U for which f p U q evadesa real half-plane, regardless of the choice of p λ n q n “ . We note that the openness of f p U q for the generic f is a trivial consequence of the open mapping theorem in eachof the cases, as the generic f is non-constant. This implies that the real task in these questions is proving the densityof the images.We fix some notation for the rest of the paper. Throughout, D r “ t| z | ă r u for general disks centered at the origin,while its boundary is denoted by S r . (That is, D “ D , S “ S .) We use the notation proj n p G q for the projection of G Ď p λ n q n “ P Ω to the n th coordinate of the product.For any ζ P S and any N P N , define t , u N r ζ s : “ ÿ j “ N a j ζ j : a j P t , u , a j “ + . A terminology we are going to use frequently is the following. For sets A , B Ď C and some ε ą
0, we say that A is an ε -net of B , if for any w P B , there exists some z P A such that | z ´ w | ă ε .Below we will also use the notation e p x q “ e π ix for x P C .2. P RELIMINARY STATEMENTS
Lemma 1.
For any c P C , set Λ ` c : “ t λ ` c : λ P Λ u , c Λ : “ t c λ : λ P Λ u . Set also f Λ p U q : “ ď p λ n q n P N P Ω f p λ n q p U q . We have the following.(a) Let U Ď S be open. Assume that f p λ n q p U q is dense in C generically for p λ n q n P N P Λ . Then for any ‰ c P C ,f p λ n q p U k q is also dense in C generically for p λ n q n P N P Λ where Λ : “ c Λ .(b) Let p U k q k “ be a shrinking sequence of open subsets of D such that diam p U k q Ñ and none of them accumu-lates to . Moreover, assume that all of the sets f p λ n q p U k q are dense in C generically for p λ n q n P N P Λ . Thenfor any c P C , all of the sets f p λ n q p U k q are dense in C generically for p λ n q n P N P Λ where Λ : “ Λ ` c. c) Assume that f Λ p U q evades a real half-plane for some open U Ď D with R U. Then for any c P C , the sameholds for f Λ ` c p U q .Proof. To prove (a), consider the homeomorphism h : Ω Λ Ñ Ω Λ given by p λ n q n “ ÞÑ p c λ n q n “ . Then the inducedmapping on the power series is simply a multiplication by c . Thus if f p λ n q p U q is dense in C , the same trivially holdsfor f p c λ n q p U q “ c f p λ n q p U q , that is in a residual subset of Ω Λ . The proof of (a) is complete.To prove (b) and (c), consider the homeomorphism of the same ilk between Ω Λ and Ω Λ , that is h : Ω Λ Ñ Ω Λ givenby p λ n q n “ ÞÑ p λ n ` c q n “ . It induces the following mapping on the corresponding functions:(1) f p λ n q ÞÑ f p λ n ` c q , f p λ n ` c q p z q : “ f p λ n q p z q ` c ´ z “ f p λ n q p z q ` g p z q . As for (b), let U k be a set given as in the statement, and assume that A : “ tp λ n q n P N P Ω : f p λ n q p U k q “ C u Ď Ω Λ is residual. Then its image A ` c : “ tp λ n ` c q n Ñ8 : p λ n q n P N P A u Ď Ω Λ is also residual. For p λ n q P A , f p λ n q p U k q is dense for any k . Consequently, by (1), we have that f p λ n ` c q p U k q forms adiam p g p U k qq -net of C . However, as g is continuous on U Ě U k for any k , one can easily see that diam p g p U k qq Ñ k Ñ 8 . This observation implies that f p λ n ` c q p U k q is dense in C for each k . The proof of (b) is complete.As for (c), observe that in (1), g p U q is bounded under the assumptions. Therefore if f Λ p U q is a subset of a realhalf-plane, then so is f Λ ` c p U q . The proof of (c) is complete. (cid:3) Our other lemma is used in the proof of Theorem 1 when ζ “ e p x q with some x P R z Q . Lemma 2.
Assume that p H j q j “ is a sequence of dense subsets of S. Then Ť k “ k ř j “ H j is dense in C , where k ř j “ H j denotes the Minkowski sum of H , ..., H k .Proof. The proof follows from three simple observations: ‚ for any k ě ¨˝ k ÿ j “ H j ˛‚ “ k ÿ j “ H j ; ‚ S ` S “ D ; ‚ D r ` S r “ D r ` r for 0 ă r ă r .Putting together these claims yields that ¨˝ k ÿ j “ H j ˛‚ “ D k for k ě
2. Consequently, ¨˝ ď k “ k ÿ j “ H j ˛‚ “ C clearly holds. (cid:3) Lemma 3.
Let ζ P S be different from ˘ , ˘ i , ˘ ω , ˘ ω , where i “ e p { q , ω “ e p { q . Then for any N P N , t , u N r ζ s is dense in C , i.e. t , u N r ζ s is an ε -net of C for any ε ą . This lemma is applied to cover most of the cases in the proof of Theorem 1, and if ζ is a root of unity, it requiresbasic algebraic number theory. We note that proceeding this way is not a must: in particular, the proof given for theexceptional cases can be reworked to any root of unity. Nevertheless we found that this approach is more elegant, andwe believe it provides a better understanding of the topic. roof. Let w P C and ε ą w with error smaller than ε with a finite sum of theform given in the statement. We go by cases.If ζ “ e p x q with x P R z Q , then the statement follows simply from Lemma 2. Namely, setting H : “ H : “ . . . : “ t ζ N , ζ N ` , ζ N ` , . . . u in Lemma 2, we obtain a certain z : “ ζ n ` . . . ` ζ n k such that | z ´ w | ă ε . Possibly there are repetitions among the n j ’s, but any ζ n j can be replaced with another ζ n j on the cost of an arbitrarily small error, which altogether verifiesthe claim.If ζ “ e p k { m q with ϕ p m q ą ϕ p m q stands for Euler’s totient function) and some k coprime to m , thenwe may assume that N “
0. Considering K : “ Q p ξ m q (where ξ m is a primitive m th root of unity), we see that Z r , ξ m , . . . , ξ ϕ p m q´ m s is the ring of integers O K of K (see [6, Chapter I, Proposition 10.2]). Then denoting by θ the embedding of K into the Minkowski space K b R C „ C ϕ p m q{ , θ p O K q is a complete lattice [6, Chapter I, Proposi-tion 5.2].Complete w as r w “ p w , , . . . , q P C ϕ p m q{ . Fixing any bounded fundamental parallelotope P for θ p O K q , one canfind α P O K and p “ p p , . . . , p ϕ p m q{ q P P such that r w “ θ p α q ` p . Any fundamental parallelotope can be rescaledby units, and since the first coordinate of a unit can be made arbitrarily small by Dirichlet’s unit theorem [6, Chap-ter I, Theorem 7.3], we may assume that necessarily | p | ă ε (by imposing that the first coordinate of the whole P belongs to an ε -ball about the origin). Writing the embedding K ã Ñ C given by ξ m ÞÑ ζ in the first coordinate in K b R C „ C ϕ p m q{ , the above argumentshows that for well-chosen a j P Z , 0 ď j ď ϕ p m q ´ ˇˇˇˇˇˇ w ´ ϕ p m q´ ÿ j “ a j ζ j ˇˇˇˇˇˇ ă ε . To get rid of potential negative coefficients, note that ´ ζ k “ ÿ ď j ď m ´ j ‰ k ζ j switches every negative coefficient to a sum of positive ones. Finally, repetitions can be treated via ζ k “ ζ k ` m , andthe proof is complete. (cid:3) Lemma 4.
Let ζ P S be different from ˘ . Then t , u N r ζ s is a -net of C for any N P N .Proof. If ζ ‰ ˘ i , ˘ ω , ˘ ω , then the statement is obvious from Lemma 3. Otherwise, we may assume N “
0, and if ζ “ ˘ i (resp. ζ “ ˘ ω or ζ “ ˘ ω ), then Z r ζ s Ă C is the lattice of Gaussian (resp. Eulerian) integers, which areknown from elementary geometry to satisfy that for any w P C , there exist a , a P Z such that | w ´ p a ` a ζ q| ă . Again, the potientially negative coefficients can be switched to a sum of positive ones by recording ´ “ ÿ j “ ζ j , ´ ζ “ ÿ j “ ζ j , and repetitions can be treated via ζ k “ ζ k ` . (cid:3) In order to make the proof of Theorem 3 more transparent, we state two simple lemmas of geometric nature first.
Lemma 5. If Λ Ď C is not contained by a half-plane of the form t z : α ď arg z ď α ` π u , then we can find ∆ “ ∆ q , ∆ , ∆ , ... ∆ q ´ elements of Λ such that (2) 0 ă arg ∆ j ` ∆ j ă π for any ď j ď q ´ . In fact, q can be chosen to be 3 or 4. We remark that Dirichlet’s unit theorem here is a slight overkill, a little more lengthy argument via Minkowski’s lattice point theorem [6,Chapter I, Theorem 4.4] would also work. roof. Multiplying Λ by a nonzero scalar does not change the condition, nor the implication. Consequently, we canassume 1 P Λ . Let ∆ “
1. By the condition on Λ , t λ : Im λ ą u is nonempty. Consequently, we can define β : “ sup λ P Λ , Im λ ą arg λ . Now by the same argument, t λ P Λ : β ă arg d ă β ` π u is nonempty. However, due to the definition of β , we candeduce that t λ P Λ : π ď arg d ă β ` π u is nonempty actually. Let ∆ be an element of it. Due to the definition of β ,we can find ∆ such that (2) is satisfied for j “ ,
1. Now if arg ∆ ą π , it is also satisfied for j “ ∆ to be any element of the necessarily nonempty t λ P Λ , Im λ ă u , which yields (2) for j “ , (cid:3) Lemma 6.
Assume that Λ Ď C is not contained by a half-plane of the form t z : α ď arg z ď α ` π u . Then there existsan appropriate R ą with the property that for any z P C satisfying | z | ą R, we can find λ P Λ such that | z ` λ | ă | z | .Proof. Fix ∆ “ ∆ q , ∆ , . . . , ∆ q ´ as guaranteed by Lemma 5. Let α : “ max ď j ď q ´ arg ∆ j ` ∆ j ă π . Now if 0 ‰ z P D R is arbitrary, we can find 0 ď j ď q ´ ∆ j we have π ` α ď arg ∆ j z ă π ´ α . Conse-quently, if we consider the triangle determined by 0 , z , z ` ∆ j , we have that the angle at z is smaller than the right angle,and its size is bounded away from π by some positive quantity. As the set of all ∆ j s is bounded, this implies that if R is large enough, then the side r , z s of the triangle is larger than the side r , z ` ∆ j s . Thus taking such an R proves thelemma. (cid:3) This lemma is used to prove the following simple but important observation.
Lemma 7.
Assume that Λ Ď C is not contained by a half-plane of the form t z : α ď arg z ď α ` π u . Then there existsR ˚ ą such that for any z, | z | ă and ď n ă n ă . . . , there exists p λ n j q j “ , λ n j P Λ such that ˇˇˇř j “ λ n j z n j ˇˇˇ ď R ˚ .Proof. An appropriate choice of R ˚ will be defined in terms of R guaranteed by Lemma 6. Notably, we will prove that R ˚ “ R ` sup λ P Λ | λ | is sufficient for the purposes of the lemma.For z “
0, the claim is trivial, regardless of the choice of p λ n j q j “ . Hence fix z ‰ | z | ă
1. The proof depends on arecursive construction of the sequence p λ n j q . Notably, let λ n P Λ be arbitrary. Now if λ n , . . . , λ n k are already definedand ˇˇˇˇˇˇ k ÿ j “ λ n j z n j ˇˇˇˇˇˇ ă R , then λ n k ` P Λ can be chosen arbitrarily as well. Otherwise we apply Lemma 6 to z n k ` Λ to define λ n k ` such that ˇˇˇˇˇˇ k ` ÿ j “ λ n j z n j ˇˇˇˇˇˇ ă ˇˇˇˇˇˇ k ÿ j “ λ n j z n j ˇˇˇˇˇˇ . These choices obviously guarantee that k ÿ j “ λ n j z n j ď R ` max λ P Λ | λ | “ R ˚ , regardless of the value of k . Consequently, the same bound holds for the sum of the series as well. (cid:3)
3. P
ROOF OF T HEOREM , P Λ . For any fixed w P C and ε ą
0, we introduce A w , ε : “ tp λ n q n P N : there exists some τ P U such that | f p λ n q p τ q ´ w | ă ε u . Fixing w P C and ε ą
0, we introduce the abbreviation A : “ A w , ε , and prove below that it is open and dense in Ω . o see that A is open in Ω , let p λ n q n P N P A , i.e. for some τ P U , ε : “ | f p λ n q p τ q ´ w | ă ε . Let N be large enough tosatisfy that ÿ n ą N sup t| λ | : λ P Λ u τ n ă ε ´ ε . Also, choose δ ą | λ n ´ λ n | ă δ for all n ď N , then ˇˇˇˇˇ ÿ n ď N λ n τ n ´ ÿ n ď N λ n τ n ˇˇˇˇˇ ă ε ´ ε . Clearly, if p λ n q n P N P ą n ď N t λ n : | λ n ´ λ n | ă δ u ˆ ą n ą N Λ Ď Ω , then | f p λ n q p τ q ´ w | ă ε , hence p λ n q n P N P A , which shows that A is open.Now we prove that A is dense in Ω . It suffices to show that A intersects any set of the form G : “ t λ u ˆ . . . ˆ t λ N u ˆ ą n ą N Λ Ď Ω . Let us fix some ˘ ‰ ζ P S X B U throughout the proof. Our goal is to find an element p λ n q n P N P G and some τ P U such that | f p λ n q p τ q ´ w | ă ε . We immediately prescribe | τ ´ ζ | ă δ with 0 ă δ ă { ˇˇˇˇˇ ÿ n ď N λ n τ n ´ ÿ n ď N λ n ζ n ˇˇˇˇˇ ă ε | τ ´ ζ | ă δ .Relabeling our original w (shifting it by ´ ř n ď N λ n ζ n ), and rescaling ε , we have to find a sequence p λ n q n ą N Pt , u n ą N and some τ P U in such a way that(3) ˇˇˇˇˇ ÿ n ą N λ n τ n ´ w ˇˇˇˇˇ ă ε , | τ ´ ζ | ă δ . Now we go by cases according to Lemmata 3–4 about the nature of ζ . To avoid notational difficulties, assume that w ‰ ζ ‰ ˘ i , ˘ ω , ˘ ω , then by Lemma 3, we may find and fix a finite sequence p λ n q N ă n ă K satisfying ˇˇˇˇˇ ÿ N ă n ă K λ n ζ n ´ w ˇˇˇˇˇ ă ε , λ n P t , u , and then if | τ ´ ζ | is small enough (consistently with the earlier prescribed | τ ´ ζ | ă δ ), ˇˇˇˇˇ ÿ N ă n ă K λ n τ n ´ ÿ N ă n ă K λ n ζ n ˇˇˇˇˇ ă ε , | τ ´ ζ | ă δ , which, setting λ K : “ λ K ` : “ . . . : “
0, together clearly imply (3).Now assume that ζ P t˘ , ˘ ω , ˘ ω u . Fix a finite set W which is a 1-net of D | w |{ ε . By Lemma 4, for any w P W ,we may find and fix a finite sequence p λ n q N ă n ă K p w q satisfying ˇˇˇˇˇ ÿ N ă n ă K λ n p w q ζ n ´ w ˇˇˇˇˇ ă , λ n p w q P t , u , where the upper bound K on the coefficient indices is uniform over w P W (this can be achieved, since W is finite).Now if | τ ´ ζ | is small enough (consistently with the earlier prescribed | τ ´ ζ | ă δ ), then ˇˇˇˇˇ ÿ N ă n ă K λ n p w q τ n ´ ÿ N ă n ă K λ n p w q ζ n ˇˇˇˇˇ ă w P W , | τ ´ ζ | ă δ . ix now M in such a way that ε { ă | τ M | ă ε { δ ă { | τ | ą {
5, and that τ has apower in the indicated annulus). Then N ă n ă K λ n p w q τ n ` M : w P W + gives rise to an ε -net of D | w | . In particular, choosing the appropriate w (for which the sum in the last display is closestto w ), and setting λ n : “ $’&’% λ n , if n ď N , λ n ´ M p w q , if N ` M ă n ă N ` K ,0 otherwise,(3) is achieved.To sum up, any A w , ε is open and dense, which in turn implies that the set č w P Q ` Q i č k “ A w , { k is residual, hence the proof of Theorem 1 is complete, at least, when 0 , P Λ . This, however, immediately gives riseto the general case by applying Lemma 1 (a)–(b), noting that in the proof of the density of A above guarantees τ ’sarbitrarily close to ζ , which means that the condition of Lemma 1 (b) is satisfied.A fairly straightforward consequence of Theorem 1 is the following: Corollary 1.
For the generic p λ n q n “ P Ω , for any ζ P S and w P C there exists p ζ k q k “ Ď D with ζ k Ñ ζ such thatf p ζ k q Ñ w. The proof is left as a simple exercise to the reader, it suffices to rely on the case of irrational arguments. With aslightly different formulation, proving this statement was Problem 9 at the prestigious Mikl´os Schweitzer MemorialCompetition for Hungarian university students in 2020, proposed by the authors. Complete solutions were given byM´arton Borb´enyi and Attila G´asp´ar, who were awarded the two first prizes of the contest, and to whom we congratulatehereby. A direct solution is available at [7] in Hungarian.4. P
ROOF OF T HEOREM f p U q holds generically.Due to Lemma 1 (a)–(b), we can assume 0 , P Λ . First we will consider the case when Λ is contained by a realline. Following our assumption, it means Λ Ď R .We will define U “ Ť k “ U k where U k “ " z : ´ k ´ k ă ℜ z ă , π ´ α k ă arg z ă π ` α k * for α k ą α k is small enough to guarantee that U k Ď S .As each U k is open, the same holds for U . Now consider any z P U k . As U k Ď S , we can choose N large enough tohave(4) ÿ n “ N z n ă . We choose α k based on the choice of N such that2 N α k ă arcsin ˆ N ˙ . This clearly implies that ´ arcsin ˆ N ˙ ă arg ˜ N ´ ÿ n “ z n ¸ ă arcsin ˆ N ˙ and π ´ arcsin ˆ N ˙ ă arg ˜ N ´ ÿ n “ z n ` ¸ ă π ` arcsin ˆ N ˙ . owever, the absolute value of each of this partial sums is at most N , which yields that their imaginary part is at most1. Consequently, Im ˜ N ´ ÿ n “ z n ¸ ď . However, it yields that for any p λ n q n “ we haveIm ˜ N ´ ÿ n “ λ n z n ¸ ď λ P Λ | λ | . Taking (4) into consideration implies Im ˜ ÿ n “ λ n z n ¸ ď λ P Λ | λ | . As it holds for any k and z P U k , we have it for any z P U , which concludes the proof of the first part.In the other direction, our argument will be similar to the one we have given in the proof of Theorem 1. Followingthe notation introduced over there, let A “ A w , ε : “ tp λ n q n P N : there exists some τ P U such that | f p λ n q p τ q ´ w | ă ε u . We will prove below that it is open and dense in Ω for any w P C and ε ą
0, which will be sufficient as previously. Asthe proof of openness can be copied verbatim, we might focus on density.It suffices to show that A intersects any set of the form G : “ t λ u ˆ . . . ˆ t λ N u ˆ ą n ą N Λ Ď Ω . Our goal is to find an element p λ n q n P N P G and some τ P U such that | f p λ n q p τ q ´ w | ă ε . We immediately prescribe | τ ´ p´ q| ă δ with δ ą ˇˇˇˇˇ ÿ n ď N λ n τ n ´ ÿ n ď N λ n p´ q n ˇˇˇˇˇ ă ε | τ ´ p´ q| ă δ .Relabeling our original w (shifting it by ´ ř n ď N λ n p´ q n ), and rescaling ε , we have to find a sequence p λ n q n ą N Pt , u n ą N and some τ P U in such a way that ˇˇˇˇˇ ÿ n ą N λ n τ n ´ w ˇˇˇˇˇ ă ε , | τ ´ p´ q| ă δ . Fix an element λ P Λ with non-vanishing imaginary part: its existence is guaranteed by the assumption that 0 , P Λ and Λ Ę R . Consider the lattice t a ` b λ : a , b P Z u . It obviously gives a δ -net of C for some δ ą
0. Consequently, t ξ p a ` b λ q : a , b P Z u . gives an ε { C for | ξ | “ ε , if ε ą ε accordingly, and fix R such that | w | ă ε R . Nowit is clear that one can find m R such that for C m R “ t a ` b λ : a , b P Z , | a | , | b | ă m R u , ξ C m R gives an ε { D R for | ξ | “ ε .Now let us notice that for large enough M R , any element of C m R can be written in the form λ k ÿ j “ p´ q n j ` l ÿ j “ p´ q n j , here k ą
0, the exponents used are pairwise distinct, and N ă p n j q kj “ , p n j q lj “ ă M R . Denote the set of such sumsby C p´ q , and motivated by this, let C p z q “ $&% λ k ÿ j “ p z q n j ` l ÿ j “ p z q n j : k ą , N ă p n j q kj “ , p n j q lj “ ă M R are distinct ,.- . As C p z q is determined by finitely many continuous functions of z , if |p´ q ´ τ | is small enough, ξ C p τ q gives a ε -netof S ε R for any ξ , | ξ | “ ε . On the other hand, we can choose τ in any neighborhood of ´ | τ | M “ ε for some M ą
0. Consequently, we have that τ M C p τ q forms a ε -net of S ε R .By definition, this implies that there exist pairwise distinct numbers N ă p n j q kj “ , p n j q lj “ ă M R such that ˇˇˇˇˇˇ¨˝ λ k ÿ j “ τ n j ` M ` l ÿ j “ τ n j ` M ˛‚ ´ w ˇˇˇˇˇˇ ă ε . Now if we define p λ n q n “ N ` such that λ n “ λ if and only if n “ n j ` M for some 1 ď j ď k , moreover, λ n “ n “ n j ` M for some 1 ď j ď l , and otherwise λ n “
0, then we immediately obtain | f p λ n q p τ q ´ w | ă ε , whichconcludes the proof. 5. P ROOF OF T HEOREM f p U q holds generically.First we will consider the case when Λ is contained by a half-plane of the form t z : α ď arg z ď α ` π u . Due toLemma 1 (a), we can assume that this half-plane is t ℜ z ě u .We will define U “ Ť k “ U k where U k “ " z : 0 ă ℜ z ă k ´ k , ´ α k ă arg z ă α k * for α k ą α k is small enough to guarantee that U k Ď S .From this point, the proof of this part is basically a simplified version of the proof of the same part of the proofof Theorem 2. Notably, for z P U k we can find a threshold index such that the tail sum is very small due to | z | beingbounded away from 1, and then by choosing α k to be small enough, we can control the argument of the precedingterms. (The relative simplicity in this case is due to the fact these arguments are all near 0, instead of being near to0 and π alternatingly.) Based on these estimates, the real part of f p λ n q p z q can be bounded from below, regardless of p λ n q n “ and z U k , which concludes the proof of the first part.Let us continue with the other part. Following the notation introduced in the proof of Theorem 1, let A “ A w , ε : “ tp λ n q n P N : there exists some τ P U such that | f p λ n q p τ q ´ w | ă ε u . We will prove below that it is open and dense in Ω for any w P C and ε ą
0, which will be sufficient as previously. Asthe proof of openness can be copied verbatim, we might focus on density. It suffices to show that A intersects any setof the form G : “ t λ u ˆ . . . ˆ t λ N u ˆ ą n ą N Λ Ď Ω . Our goal is to find an element p λ n q n P N P G and some τ P U such that | f p λ n q p τ q ´ w | ă ε . We immediately prescribe | τ ´ | ă δ with δ ą ˇˇˇˇˇ ÿ n ď N λ n τ n ´ ÿ n ď N λ n n ˇˇˇˇˇ ă ε | τ ´ | ă δ . As previously, relabeling our original w (shifting it by ´ ř n ď N λ n n ), and rescaling ε ,we have to find a sequence p λ n q n ą N P Λ n ą N and some τ P U in such a way that ˇˇˇˇˇ ÿ n ą N λ n τ n ´ w ˇˇˇˇˇ ă ε , | τ ´ | ă δ . Define ∆ , ∆ , . . . , ∆ q ´ P Λ as guaranteed by Lemma 5. Denote their set by V , and define the convex polygon P “ conv p V q . Due to the choice of V , 0 P int p P q , that is D r Ď V for small enough r . otice that if P has diameter δ , then V is clearly a δ -net of P . Moreover, for any m we have that the Minkowski sum ř mi “ V is a δ -net of ř mi “ P : the proof proceeds by induction, capitalizing on the simple observation that ř mi “ P “ V ` ř m ´ i “ P . It clearly yields that ř mi “ V is a δ -net of D mr as well for any m . Consequently, ξ ¨ ř mi “ V gives an ε { D ε mr for any m and for | ξ | “ ε , if ε ą ε accordingly, noting that it does not dependon m . Now fix R ˚ as guaranteed by Lemma 7, and based on the choice of R ˚ and ε , fix m such that | w | ` R ˚ ă ε mr .Let us remark that any element of ř mi “ V is expressible in the seemingly complicated form q ´ ÿ j “ ∆ j k j ÿ l “ n p j q l , where each k j ą
0, and the exponents used are pairwise distinct and their union equals t N ` , N ` , ..., N ` m u . Letus denote the set of such combinations by C p q , and motivated by this, let C p z q “ $&% q ´ ÿ j “ ∆ j k j ÿ l “ z n p j q l : k ą , N ă p n p j q l q k j l “ ď N ` m are distinct and their union is t N ` , N ` , ..., N ` m u ,.- . As C p z q is determined by finitely many continuous functions of z , if | ´ τ | is small enough, ξ C p τ q gives a ε -net of D ε mr for any ξ , | ξ | “ ε . On the other hand, we can choose τ in any neighborhood of 1 so that | τ | M “ ε for some M ą
0. Consequently, we have that τ M C p τ q forms an ε -net of D ε mr .So far the coefficients of the power series we would like to define are fixed in the places p i q Ni “ . Motivated bythe previous paragraph, we would like to set aside the places p N ` i ` M q mi “ . Notably, these are the places whichare intimately connected to the lastly defined τ M C p τ q . Consequently, we apply Lemma 7 at this point for τ and thecomplementary sequence p N ` , N ` , ..., N ` M , N ` m ` M ` , N ` m ` M ` , ... q : we can find elements of Λ corresponding to these places, p λ n q N ` Mn “ N ` , p λ n q n “ N ` m ` M ` such that w “ N ` M ÿ n “ N ` λ n z n ` ÿ n “ N ` m ` M ` λ n z n ď R ˚ . Consequently, | w ´ w | ď | w | ` R ˚ ă ε R . This implies that there exist pairwise distinct numbers N ă ´ p n p j q l q k j l “ ¯ q ´ j “ ď N ` m , such that their union fills t N ` , N ` , ..., N ` m u , and(5) ˇˇˇˇˇˇ q ´ ÿ j “ ∆ j k j ÿ l “ τ n p j q l ` M ´ p w ´ w q ˇˇˇˇˇˇ ă ε . What remains from the definition of p λ n q is fixing p λ n q N ` m ` Mn “ N ` ` M , which we carry out now based on (5). Notably let λ n “ ∆ j if and only if n “ n p j q l ` M for some 1 ď l ď k j . Then by the definition of w we obtain | ř n “ N ` λ n τ n ´ w | ă ε .It concludes the proof. 6. C ONCLUDING REMARKS
Even though with a careful separation of cases we managed to generalize the most natural result given by Theorem1, our results are far from being complete. In our view, the most interesting open problem related to them is whether f p U q “ C holds generically in the setup of our theorems in fact, similarly to what is proved in [1]. As f is uniformlylocally bounded in D , we clearly cannot rely on techniques similar to the ones seen there, hence answering this questionrequires additional ideas.Another interesting aspect partially inspired by this paper is that whether we can rearrange the quantifiers in ourstatements to some extent. More explicitly, each of our theorems addresses the question of what the generic image isof a fixed open set. It would be desirable to find extensions of this result, for example a nontrivial family of open setssuch that generically, the image of each of them is dense. EFERENCES[1] J.-P. Kahane,
Baire’s category theorem and trigonometric series , J. Anal. Math. , (2000), 143–182.[2] J.-P. Kahane, Some Random Series of Functions , Heath, Mass., 1968; 2nd edn. Cambridge Univ. Press, 1985, (1993).[3] S. Kierst, E. Szpilrajn,
Sur certaines singularit´es desfonctions analytiques uniformes , Fund. Math. , (1933), 267-294.[4] K. Kuratowski, Topologie , Vol. 1, 4th ed., PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, (1966).[5] B. Maga, P. Maga,
Random power series near the endpoint of the convergence interval , Publ. Math. Debrecen, (3-4). (2018), 413-424.ISSN 0033-3883[6] J. Neukirch, Algebraic number theory , Grundlehren der Mathematischen Wissenschaften, Vol. 322, Springer-Verlag, Berlin, 1999.[7]
E ¨
OTV ¨ OS L OR ´ AND U NIVERSITY , D
EPARTMENT OF A NALYSIS , P ´
AZM ´ ANY
P ´
ETER S ´ ET ´ ANY
UDAPEST , H–1117 H
UNGARY
Email address : [email protected] MTA A
LFR ´ ED R ´
ENYI I NSTITUTE OF M ATHEMATICS , POB 127, B
UDAPEST
H-1364, H
UNGARY
Email address : [email protected]@gmail.com