Real Mahler Series

During the 1920’s Mahler was able to show that the function

takes on transcendental values for every algebraic number $0<x<1$ [11]. Since then his work has been generalized and the Mahler Method is now a staple of transcendence theory [13]. Further work by Cobham has highlighted connections to automata theory [7] [3]. Cobham was able to prove the following.

The proof is infamous for its difficulty despite using only elementary techniques. The correspondence between $k$-automatic sequences and what are known as Mahler functions has been widely known for at least 15 years [8]. Cobham’s theorem naturally leads to the analogous question for Mahler functions, which we will now introduce. The astute reader will immediately verify that the series in equation 1 satisfies the below functional equation:

We say that particular $F(x)$ is $2$-Mahler of degree $1$. Generalizing this notion, suppose a series satisfies

We will then say that $F$ is $k$-Mahler of degree $d$. Here $P_{i}$ and $A$ are polynomials. An equivalent formulation would be to say that

forms a $d$-dimensional vector space over $K[x]$.

Let $\alpha>0$ and $\beta>0$ be real numbers. The reader will recall that if $\frac{\log(\alpha)}{\log(\beta)}\not\in\mathbb{Q}$ then $\alpha$ and $\beta$ are said to be multiplicatively independent. Loxton and Van der Poorten conjectured the following counterpart of Cobham’s theorem for Mahler functions

Notice that the converse of the theorem holds trivially, rational functions are $\alpha$-Mahler for any $\alpha$. In 2013 this conjecture was resolved by Bell and Adamczewski who were the first to prove the more general theorem, over any field of characteristic $0$ [1] .

The theorem was also proven by Schafke and Singer through entirely different methods in 2017 [15]. In this paper we will extend the result beyond positive integers. The main results are as follows.

Here we outline the structure of the argument giving a summary of each section.

1. 1.

   In the next section we will define and summarize some basic facts about Hahn series. We then go on to prove a few quick lemmas that allow us to reduce the form of the Mahler functional equations to something more manageable, at least in the rational case. Later on in the proof we will use some more reductions of a similar nature that will be proven as we need them.

2. 2.

   Afterward, we will restrict our vision for a moment, and look at only Laurent series instead of Hahn series. The goal of this section is to prove a nonexistence result of $\alpha$-Mahler Laurent series when $\alpha$ is not an integer. This result, while somewhat interesting in its own right, will allow us to make further claims when dealing with the more general case of Hahn series.

3. 3.

   We will analyze the supports of Hahn series under the Mahler functional equations to gain a broad understanding of how they interact. This will then prove invaluable in making further reductions to a more tractable problem.

4. 4.

   Finally, we will tackle the problem and prove theorem 1.4 under the assumption that $\alpha$ and $\beta$ are rational. In the case that $\alpha$ and $\beta$ are integers we appeal to the Adamczewski-Bell theorem. In the other case we aim to reduce the Hahn series by putting together all the results found in the previous sections.

5. 5.

   Some additional details are needed when one of $\alpha$ or $\beta$ is irrational. We delineate these cases and prove theorem 1.5 for them. This completes the paper.

In this section we introduce some definitions and notation. We will also take the time to prove some elementary facts and reductions that will be helpful to us later on. Let $K$ be a number field with $K[x],K(x),K[[x]],$ and $K[[x]][x^{-1}]$ denoting polynomials, rational functions, power series and Laurent series over $K$. We will use $(p,q)$ as a shorthand for the greatest common denominator of integers $p$ and $q$.

Let $\alpha=\frac{p}{q}$ be a non-integer positive rational with $\left(p,q\right)=1$. Require that $\alpha\neq\frac{1}{n}$ for an integers $n$. Then we will show here that $F$ is a $\alpha$-Mahler Laurent series if and only if $F$ is rational. Notice that the backward direction is trivial. The restriction on $\alpha$ is quite necessary. If $n$, a natural number, is multiplicatively dependent to $\frac{1}{n}$ we can construct $\frac{1}{n^{k}}$-Mahler functions from $n$-Mahler functions. We give one such example and leave the rest to the reader.

Let $F(x)=\sum_{n=2}^{\infty}x^{2^{n}}$. Then $F(x)$ is $2$-Mahler and satisfies

Normalize so that we get a homogeneous functional equation

And applying the $x\mapsto x^{\frac{1}{2}}$ operator twice:

yields a $\frac{1}{2}$-Mahler functional equation.

Moving on to a proof, we say a Laurent series, $F$, in $K[[x]][x^{-1}]$ is $\alpha$-Mahler of degree $d$ if there exist polynomials $P_{0}(x),\ldots,P_{d}(x)$ in $K[x]$ so that

under these conditions we must have

The approach is to first reduce to the degree $1$ case. From there we argue using an infinite product formulation of the problem.

We use $\mathbb{Z}+\gamma$ to denote the set $\{z+\gamma\}$ for $z\in\mathbb{Z}$.

We will later use the more abstract version of this notation. If $G$ is a Hahn series we will take ${}_{A}[G]$ to mean the Hahn series containing only terms with powers contained in $A$. An immediate consequence of this definition is $P({}_{A}[G])=A\cap P(G)$.

We are ready to prove the central result of this section and let $\alpha>1$. Write $F(x)=x^{v(F)-1}G(x)$, then $G(x^{l})$ is also $\alpha$-Mahler for some $l$ that divides $v(Q_{i})$ for polynomials in the functional equation defining If we know

then we know the powers of $x$ across the left hand side must appear in $2$ of the $d$ terms and cancel to $0$. In particular, the minimal terms in each of the $d$ summands must satisfy this constraint so we know

for some integers $b_{0}$ and $b_{1}$ and naturals $n$ and $m$. If $\alpha$ is a rational that is not an integer then

while $\frac{b_{1}-b_{0}}{l}$ is. Contradiction. ∎

Let $S$ be a well ordered subset of $\mathbb{R}$ under $<$. Consider the formal Hahn series $F\in K^{S}[[x]]$. Suppose $F$ is a homogeneous $\alpha$-Mahler function as well, with $\alpha\in\mathbb{R}$ for some $\alpha>1\in\mathbb{R}$. Again, by Lemma 3.3, $\alpha>1$ is assumed to be the case.

Let $P_{d}(x)\neq 0$ and assume $F$ satisfies

where $F(x)=\sum_{s\in S}f_{s}x^{s}$. We can assume $f_{s}$ is never $0$ by taking a subset of $S$ if need be. $S$ must be countable, and use the well-ordering principle to index $S=\{s_{0},s_{1},\ldots\}$.

The point of this section is to establish this theorem.

We use this theorem later on in the proof of our main theorem to reduce to the case where the set of powers of $x$ is a subset of $\mathbb{Z}[\alpha][\alpha^{-1}]$.

We shall soon see that we can sharpen the restriction $P(F)\subset\mathbb{Z}[\alpha][\alpha^{-1}]+\alpha^{\mathbb{Z}}s$ further still, under some assumptions. We will however, wait and do this spread out over a few cases. For now, we point out another restriction, namely

We end this section we a small observation of what occurs in the case where one of the ${}_{T(s)}[F]$ is rational, which will be useful to us.

As before let $F(x)\in K^{\mathbb{R}}[[x]]$ be a Hahn series over $\mathbb{R}$. Let $\alpha=\frac{p_{1}}{q}$ and $\beta=\frac{p_{2}}{q}$ be multiplicatively independent rationals (although not necessarily written in lowest common form). In this section we aim to prove theorem 1.4

We begin by strengthening the previous Lemma. Our goal is to show we can restrict to where $P(F)\subset\mathbb{Z}[\alpha,\alpha^{-1}]$, and similarly for $\beta$. Decompose $F$ across equivalence classes as in theorem 5.1. By focusing on each term of the decomposition individually we can restrict to where $P(F)\subset T(s)$ for some properly chosen $s$ by Lemma 5.1. That is, suppose $P(F)\subset\alpha^{\mathbb{Z}}s+\mathbb{Z}[\alpha,\alpha^{-1}]$ for some $s\not\sim 0$. We show that $s$ must be rational. In the case that $s$ is rational, we can take $F(x)$ to a large enough power by the operator $x\mapsto x^{l}$.

The case where both $\alpha$ and $\beta$ are integers is the Adamczewski-Bell theorem [1]. Therefore we can assume at least one of $\alpha$ or $\beta$ is not an integer.

We quickly deal with the case that $\beta$ is an integer. Assume that $\beta$ is an integer. By the theorem we just proved, 5.1 ,we have that $P(F)\subset\mathbb{Z}$ in which case $\alpha$ being a non-integer rational and $F(x)$ being $\alpha$-Mahler contradicts the theorem established two sections ago, theorem 4.1 unless $F(x)$ is rational.

We begin the main thrust with a Lemma.

So we consider instead $G(x)=F(x^{l})$. We haven’t yet specified $N$, and $l$ depends on it. We will leave it unspecified for now. Let $p$ be a prime and $v_{p}$, the $p$-adic valuation. We need one more quick lemma before we embark on the final stretch of the rational case.

Moving on to the proof of theorem 1.4.

Using the above Lemma we can complete the proof in a few lines. Take any prime, $p$. By construction, there is an element of the intersection, $\mathbb{Z}[\alpha^{n^{\prime}}\beta^{m^{\prime}}]$ which does not contain $\frac{1}{p}$. From this we conclude

But theorem 4.1 forbids this. Contradiction.

Some care needs to be devoted to the case where one of $\alpha$ or $\beta$ is irrational. For this section we assume $\alpha$ and $\beta$ are algebraically independent. We demonstrate the analogous theorem to theorem 1.4 in this case, theorem 1.5.

To summarize, we have shown that there exist no Laurent series that are $\alpha$-Mahler for $\alpha$ multiplicatively independent from a natural number, other than rational functions $F\in K(x)$. We then use this result to prove there exist no doubly Mahler Hahn series, $\alpha$- and $\beta$-Mahler Hahn series, where $\alpha$ and $\beta$ are multiplicatively independent rationals and algebraically independent numbers. A question open for further investigation is whether this result for irrational numbers can be relaxed to $\alpha$ and $\beta$ multiplicatively independent instead of algebraically independent:

Acknowledgements. — The author would like the thank Jason Bell for suggesting the problem and discussions surrounding the problem. Without him this paper would not be possible.