Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results:

In [1] Gross constructed an entire function for which every point in the complex plane is an asymptotic value.

In [2] Heins proved that every Suslin analytic set is an asymptotic set of some entire function.

In [3] Heins proved that for every countable set A in the complex plane there exists an entire function whose set of locally omitted values coincides with A.

**(Q1)** My question is, are any of these examples locally univalent functions?

Unfortunately I was not able to varify these things by myself since I don't read German [1] and since I was not able to find an article [2] in our library basis or online.

Regarding the third result I think that the answer is positive, since the function is obtained as an associated mapping function of a parabolic surface spread over the sphere with only logarithmic ramification points. Is that correct?

**(Q2)** Do you know any (other) references with a construction of an entire function having a prescibed singular set - closed subset of a complex plane (I've seen the paper of Bishop, Constructing entire functions by QC-folding, where this is done)

*[1] W. Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzwert ist (German). Math. Ann. 79 (1–2), 201–208 (1918)*

*[2] M. Heins, The set of asymptotic values of an entire function, Tolfte Skandinaviska Matematikerkongressen (Lund, Sweden, 1953), Proceedings of the Scandinavian Math. Congress,Lund, 1954, pp. 56–60*

*[3] M. Heins: Asymptotic spots of entire and meromorphic functions. Ann. Math. (2) 66, 430–439 (1957)*