Clarifying a definition and theorem by Adamczewski and Bugeaud? While looking at an answer on the MSE concerning the possible normality of real algebraic irrational numbers, a commentator posted the the following link to a paper by Adamczewski and Bugeaud, Annals of Mathemetics. In this paper the authors define the following:
$\underline{\text{Defintion}}$: Let $p(n)$ counts the number of distinct blocks of $n$ digits (on the alphabet $\{0,1,\ldots,b-1\}$ ) of a real number $\alpha$. We call $p(n)$ the complexity function of $\alpha$. 
The authors then go on to state the following theorem:
Theorem 1: Let $b\geq 2$ be an integer. The complexity function of the b-ary expansion of every irrational algebriac number satisfies $$\lim_{n \to \infty}\inf {p(n) \above 1.5 pt n}= + \infty$$
The authors state other theorems relating to infinite words and transcendence. 
I am embarrassed to ask but "in words" what exactly is the definition defining and what is Theorem 1 saying ? An explanation in base-10 suffices and would most likely be more intuitive for me to understand. 
 A: Here's an example.  Consider the following real number $$\alpha=0.4646424743\dots$$ and $n=3$.  The "blocks of $3$ digits of $\alpha$" are $464$, $646$, $464$, $642$, $424$, $247$, $474$, $743$, and so on.  That is, a block of $3$ digits is just a sequence of $3$ consecutive digits in the expansion of $\alpha$.  So $p(3)$ is the number of distinct such blocks which appear somewhere in the expansion of $\alpha$.  In this case we can say $p(3)\geq 7$, since we have already seen $7$ distinct such blocks (but probably $p(3)$ is larger than that since to compute $p(3)$ we need to look at all the blocks in the entire expansion).
So for any fixed number $\alpha$, we get a function $p(n)$.  Theorem 1 is stated somewhat awkwardly, but it says that whenever $\alpha$ is an irrational algebraic number, $p(n)/n$ goes to $\infty$ as $n$ goes to $\infty$.  That is, for any $K$, there exists $N$ such that $p(n)/n>K$ for all $n>N$.
(Theorem 1 states that the lim inf is $\infty$, rather than the limit, but this is equivalent.)
