Question

Hi everybody. I'm looking for an analogue of irrationality measure for formal power series with integer coefficient, the elements of $\mathbb{Z}[[x]]$. For any $f \in \mathbb{Z}[[x]]$ and positive integer $g$, I thought to define something like $$m_g(f) := \sup_{p,q} \mbox{ord}(f - p / q)$$ where $p,q \in \mathbb{Z}[x]$ satisfy $\deg p, \deg q \leq g$ and $q \neq 0$; $\mbox{ord}(h) := n_0$ for any formal Laurent series $h = \sum_{n=n_0}^\infty a_n x^n$ with $a_{n_0} \neq 0$ and $\mbox{ord}(0) := +\infty$. Note that $m_g(f)$ is finite for all $g$ if and only if $f$ is irrational, otherwise $m_g(f) = +\infty$ for $g$ sufficently large. Do you have any references on this? Thank you.

Answer 1

Going by what's in Math Reviews, these two papers might be relevant:

Sandra Delaunay, Approximation diophantienne et distances ultramétriques non standard, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 629–661, MR2188586 (2006i:11085).

Dinesh S. Thakur, Diophantine approximation in finite characteristic, in Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 757–765, Springer, Berlin, 2004, MR2037123 (2004m:11109).

Answer 2

I think you will enjoy the paper "Irrationality of Power Series for Various Number Theoretic Functions", by W.D. Banks, F. Luca and I.E. Shparlinski. They use your $m_g$ as a measure of irrationality and give asymptotics on $m_g(f)$ for a variety of different power series $f$. They focus on power series with coefficients coming from arithmetic functions such as the Euler totient function, number of (prime, squarefree...) divisors, sum of divisors, Liouville function etc.

Answer 3

Note that thanks to Pade approximation we have $m_g(f) \geq 2g + 1$ for all $g \geq 1$ and $f \in \mathbb{Z}[[x]]$.