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A note on transcendental entire functions mapping uncountable many Liouville numbers into the set of Liouville numbers.

1. Introduction. A real number [xi] is called a Liouville number, if there exist infinitely many rational numbers [([p.sub.n]/[q.sub.n]).sub.n[greater than or equal to]1], with [q.sub.n] [greater than or equal to] 1 and such that

0 < [absolute value of [xi] - [[p.sub.n]/[q.sub.n]]] < 1/[q.sup.n.sub.n].

It is well-known that the set of the Liouville numbers L is a [G.sub.[delta]]-dense set and therefore an uncountable set.

In his pioneering book, Maillet [5, Chapitre III] discusses some arithmetic properties of Liouville numbers. One of them is that, given a non-constant rational function f, with rational coefficients, if [xi] is a Liouville number, then so is f([xi]). Motivated by this fact, in 1984, as the first problem in his paper Some suggestions for further research, Mahler [4] raised the following question (this question also appeared in other texts, for example in the Bugeaud's book [3, p. 215] and in Waldschmidt's paper [10, p. 281]).

Mahler's question. Are there transcendental entire functions f(z) such that if [xi] is any Liouville number, then so is f([xi])?

He also said that: "The difficulty of this problem lies of course in the fact that the set of all Liouville numbers is non-enumerable". Alniacik [1] and Bernik and Dombrovskii [2] obtained some results related to this question. Also, recently, some authors (see [6-8]) constructed classes of Liouville numbers which are mapped into Liouville numbers by transcendental entire functions.

We remark about the existence of more specific classes of Liouville numbers in the literature, for example, the strong and semi-strong Liouville numbers (see, for instance, [9]). Here, we shall define an uncountable subclass of the strong Liouville numbers which we named as ultra-strong Liouville numbers: a real number [xi] is called an ultra-strong Liouville number, if the sequence [([p.sub.n]/[q.sub.n]).sub.n] of the convergents of its continued fraction satisfies

0 < [absolute value of [xi] - [[p.sub.n]/[q.sub.n]]] < 1/[q.sup.n.sub.n], for all n [greater than or equal to] 1.

We denote this set by L. Define a sequence A = [([a.sub.n]).sub.n] by [a.sub.1] = [a.sub.2] = [a.sub.3] = 1 and [a.sub.j] [member of] {[v.sub.j-1], [v.sub.j-1] + 1}, for j [greater than or equal to] 4, where [mathematical expression not reproducible]. Then the number [[xi].sub.A] := [0, [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4], ...] is an ultra-strong Liouville number. In fact, if [0, [a.sub.1], ..., [a.sub.n]] = [p.sub.n]/[q.sub.n], then, by construction, [mathematical expression not reproducible] (here, we used the well-known inequality ([a.sub.1] + 1) ... ([a.sub.n] + 1) > [q.sub.n]). Thus

[mathematical expression not reproducible]

as desired. The set L is uncountable because there exists a binary tree of possibilities for [[xi].sub.A], since we have two possibilities for [a.sub.k] in each step (k [greater than or equal to] 4).

We remark that the set L is different from all previously constructed sets in [6-8] since all those sets are [G.sub.[delta]]-dense while L does not have this property (in fact, it was proved in [9] that the sum of two strong Liouville numbers is a Liouville number). Moreover, L is not a subset of those sets. In fact, roughly speaking, the sets constructed in [6,7] are Liouville numbers satisfying, in particular, that [mathematical expression not reproducible], for an infinite sequence of rational numbers [([p.sub.n]/[q.sub.n]).sub.n[greater than or equal to]1]. Thus, we contruct [xi] recursively with the property that [a.sub.n+1] = [q.sup.n-2.sub.n] (note that the construction of [q.sub.n] only depends on [a.sub.j], for 0 [less than or equal to] j [less than or equal to] n). In this case, clearly [xi] [member of] L, but if [mathematical expression not reproducible] (by reordering indexes if necessary), then we arrive at an absurdity as [mathematical expression not reproducible], for all sufficiently large n.

In this paper, we prove the following result:

Theorem 1.1. Let [([s.sub.n]).sub.n[greater than or equal to]1] be a sequence of positive integers satisfying that, for any given k [greater than or equal to] 1, the quotient [s.sub.n]/[s.sup.k.sub.n-1] tends to infinity as n [right arrow] [infinity]. Let F : C [right arrow] C be a function defined by

[mathematical expression not reproducible],

where [[alpha].sub.k] = 1 if k = [s.sub.j] and [[alpha].sub.k] = 0 otherwise. Then F is a transcendental entire function such that F(L) [subset or equal to] L. In particular, there exist uncountable many transcendental entire functions taking the set of the ultra-strong Liouville numbers into the set of Liouville numbers.

Let us describe in a few words the main ideas for proving Theorem 1.1. First, our desired function has the form [mathematical expression not reproducible], where [([t.sub.n]).sub.n] is an integer sequence with a very fast growth. We then approximate F([xi]), where [xi] is an ultra-strong Liouville number, by a convenient truncation [F.sub.m]([p.sub.n]/[q.sub.n]) for sufficiently large m and n. After that, we take the advantage of the fact that our series has much more zero coefficients than a strongly lacunary series. This, together with the fact that well-approximations come from the continued fraction, allows us to arrive at our desired estimate. The proof splits in two cases depending on the growth of the denominator of the convergents of [xi].

2. The proof of Theorem 1.1. Let [([s.sub.n]).sub.n[greater than or equal to]1] and F(z) be defined as in the statement of Theorem 1.1. Clearly, F is a transcendental entire function and now, we shall prove that F(L) [subset or equal to] L.

Let [xi] be an ultra-strong Liouville number and let [([p.sub.n]/[q.sub.n]).sub.n[greater than or equal to]1] be the sequence of the convergents of its continued fraction. This means that 0 < [absolute value of [xi] - [[p.sub.n]/[q.sub.n]]] < 1/[q.sup.n.sub.n], for all n > 1. Set [[phi].sub.n] = [[phi].sub.n]([xi]) as the smallest positive integer k such that [q.sub.n] [less than or equal to] [10.sup.k!].

We have two cases to consider:

Case 1. When [[phi].sub.n] [less than or equal to] [n.sup.k] for some k [greater than or equal to] 1 and all n [greater than or equal to] 1.

In this case, [mathematical expression not reproducible]. Now, consider the truncations

[mathematical expression not reproducible],

and the convergents

[mathematical expression not reproducible].

Note that [mathematical expression not reproducible] (where den(z) denotes the denominator of a rational number z). We shall prove that F([xi]) is well-approximated for the rational numbers 7n in a convenient way which ensure that it is a Liouville number. Since [mathematical expression not reproducible], we need to estimate each part in the right-hand side. For that, we have

[mathematical expression not reproducible],

and it holds that

[mathematical expression not reproducible],

since max{[absolute value of [xi]], [absolute value of [p.sub.n]/[q.sub.n]]} < 1 + [absolute value of [xi]] (for all sufficiently large n). Then

[mathematical expression not reproducible],

where we used that [[summation].sub.k[greater than or equal to]1] k/[10.sup.k!] = 0.1200030.... Since [q.sub.m] > [2.sup.(m-4)!], for m [greater than or equal to] 5 (here we used that [mathematical expression not reproducible] and so [q.sub.n+1] [greater than or equal to] [q.sup.n-1.sub.n] yielding, recursively, that [q.sub.n+1] > [2.sup.(n-3)!]), then

[mathematical expression not reproducible],

for all sufficiently large n. Therefore, a straightforward calculation gives

(1) [absolute value of [F.sub.n]([xi]) - [[gamma].sub.n]] < 1/[(den([[gamma].sub.n])).sup.n],

for all sufficiently large n.

Now, for estimating [absolute value of F([xi]) - [F.sub.n]([xi])], we shall consider the truncation in n = [s.sub.j-1] satisfying [mathematical expression not reproducible]. Thus, we have

(2) [mathematical expression not reproducible],

where we used that [mathematical expression not reproducible] since [s.sup.4(k+2).sub.j-1] < [s.sub.j] for all sufficiently large j and [mathematical expression not reproducible]. By combining (1) and (2), we obtain

[absolute value of F([xi]) - [[gamma].sub.n]] < 2/[(den([[gamma].sub.n])).sup.n],

for all sufficiently large n. Thus, in order to prove that F([xi]) is a Liouville number, it suffices to prove that [absolute value of F([xi]) - [[gamma].sub.n]] > 0 for infinitely many integers n. Suppose the contrary, then [[gamma].sub.n] = p/q for all sufficiently large integers n. By multiplying this equality by [mathematical expression not reproducible], we get that [mathematical expression not reproducible] divides q for infinitely many integers n which are absurds. Thus F([xi]) is a Liouville number as desired.

Case 2. When [[phi].sub.n] is not bounded for [n.sup.k] for all k [greater than or equal to] 1.

In this case, we have the existence of infinitely many pairs ([n.sub.j], [k.sub.j]) [member of] [Z.sup.2.sub.[greater than or equal to]1] such that

[mathematical expression not reproducible].

Now, define [t.sub.j] as the smallest integer such that [mathematical expression not reproducible] and define our approximants as

[mathematical expression not reproducible].

Note that [mathematical expression not reproducible].

As before, we want to obtain an estimate for [mathematical expression not reproducible]. First, we shall estimate [mathematical expression not reproducible]. For that, note that for all sufficiently large j, we have [mathematical expression not reproducible] and then

[mathematical expression not reproducible],

where we used that [mathematical expression not reproducible], for all sufficiently large j. Then, we have

(3) [mathematical expression not reproducible].

Now, we shall estimate [mathematical expression not reproducible]. For that, we have

[mathematical expression not reproducible].

As in the previous case, we get

[mathematical expression not reproducible],

since [mathematical expression not reproducible]. Then

[mathematical expression not reproducible].

Now, we use the well-known fact that

[mathematical expression not reproducible].

Also, by definition, since [mathematical expression not reproducible], then [mathematical expression not reproducible]. Therefore, for all [k.sub.j] [greater than or equal to] 5, we have

[mathematical expression not reproducible].

Therefore

(4) [mathematical expression not reproducible].

Note that [mathematical expression not reproducible], since [mathematical expression not reproducible]. However, we have the inequality

[mathematical expression not reproducible],

since [mathematical expression not reproducible]. The above inequality combined with (4) gives

(5) [mathematical expression not reproducible].

By combining (3) and (5) we obtain

[mathematical expression not reproducible],

for all sufficiently large j. Since [absolute value of F([xi]) - [[gamma].sub.j]] > 0 for all sufficiently large j (by a same argument as before), then F([xi]) is a Liouville number as desired. In conclusion, F(L) [subset or equal to] L.

The proof of the existence of the uncountable many functions F(z) with this property completes because there is a binary tree of different possibilities for [([s.sub.n]).sub.n[greater than or equal to]1] (and each choice defines a different function F(z)). For example, take [s.sub.n] = [a.sup.n!.sub.n], where [a.sub.n] [member of] {2, 3}.

doi: 10.3792/pjaa.93.111

Acknowledgements. The authors are grateful to the referee for some suggestions which improved the quality of this work. They also thank CNPq-Brazil for the financial support.

References

[1] K. Alniacik, The points on curves whose coordinates are U-numbers, Rend. Mat. Appl. (7) 18 (1998), no. 4, 649-653.

[2] V. I. Bernik and I. V. Dombrovskii, [U.sub.3]-numbers on curves in [R.sup.2], Vestsi Akad. Navuk Belarusi Ser. Fiz. Mat. Navuk 1992, no. 3-4, 3-7, 123.

[3] Y. Bugeaud, Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, Cambridge, 2004.

[4] K. Mahler, Some suggestions for further research, Bull. Austral. Math. Soc. 29 (1984), no. 1, 101-108.

[5] E. Maillet, Introduction a la Theorie des Nombres Transcendants et des Proprietes Arithmetiques des Fonctions, Gauthier-Villars, Paris, 1906.

[6] D. Marques and C. G. Moreira, On a variant of a question proposed by K. Mahler concerning Liouville numbers, Bull. Aust. Math. Soc. 91 (2015), no. 1, 29-33.

[7] D. Marques and J. Ramirez, On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 25-28.

[8] D. Marques and J. Schleischitz, On a problem posed by Mahler, J. Aust. Math. Soc. 100 (2016), no. 1, 86-107.

[9] G. Petruska, On strong Liouville numbers, Indag. Math. (N.S.) 3 (1992), no. 2, 211-218.

[10] M. Waldschmidt, Open Diophantine problems, Mosc. Math. J. 4 (2004), no. 1, 245-305, 312.

By Jean LELIS, *) Diego MARQUES *) and Josimar RAMIREZ **)

(Communicated by Kenji FUKAYA, M.J.A., Oct. 12, 2017)

*) Departamento de Matematica, Universidade de Brasilia, Campus Universitario Darcy Ribeiro, Brasilia-DF 70910-900, Brazil.

**) Departamento de Matematica, Universidade Federal de Uberlandia, Av. Joao Naves de Avila, 2121, Bairro Santa Monica, Campus Santa Monica, Uberlandia-MG 38400-902, Brazil.
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Author:Lelis, Jean; Marques, Diego; Ramirez, Josimar
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Sep 1, 2017
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