Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.


Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving left, $1$ means moving right). It is then well-known that, in the Cantor space $2^\mathbb{N}$ under the Lebesgue measure, the set $A$ of non-recurrent walks -- i.e. those sequences $x$ for which $\frac{\sum\_{k=1}^n x(k)}{n} = \frac{1}{2}$ for only finitely many $n$ -- is null. I am curious as to the Hausdorff dimension of this set, but I do not see how to figure this. Thus my question:



> 
> What is the Hausdorff dimension of the set $A$ of non-recurrent walks?
> 


Perhaps this too is already well-known, but if so I could not locate the result. I hope this question is not trivial, though if it is at least I will have learned that. I look forward to any replies; this seems like a marvelous site!



Hausdorff dimension of non-recurrent walks
First of all the dimension of the space. You didn't give the metric you want to use, so I'll use my favourite one: two points are at distance $2^{-n}$ if they first disagree in the $n$th symbol. The 1-Hausdorff measure agrees with coin-flipping measure so that the full space has Hausdorff dimension 1.

Now for the non-recurrent subset. 
It's also of Hausdorff dimension 1. Probably the simplest way to see this is to use some technology: consider the measure $\mu\_\alpha$ that is coin-flipping with weights $\alpha$ and $1-\alpha$. Clearly for $\alpha\ne1/2$ this measure is supported on the non-recurrent set. But the Hausdorff dimension of the measure is $(-\alpha\log\alpha-(1-\alpha)\log(1-\alpha))/\log 2$. This is a lower bound for the Hausdorff dimension of the non-recurrent set. But as $\alpha\to1/2$, this lower bound converges to 1.

In case you have an aversion to dynamical systems there is a way to do this with a single (non-invariant) measure and get the result without taking any limits. Let $n\_1 < n\_2 < \ldots$ be a sequence of density 0 such that the number of terms up to $N$ ($T(N)$ say) grows faster than $\sqrt{3N\log\log N}$ (e.g. $(n\_i)$ is the sequence $\lfloor i^2/\log i\rfloor$). Now build the measure that is fair coin-tossing at each $n$ except at the $n\_i$ when you always put a 1. Since the fair part of the process puts you in the range $\pm\sqrt{(2+\epsilon)N\log\log N}$ for all sufficiently large $N$ (by the Law of the Iterated Logarithm), the unfair part of the process guarantees that you only return to 0 finitely many times. Hence this measure is supported on the non-recurrent set. This measure has Hausdorff dimension 1: the measure of a $2^{-N}$ neighbourhood of a typical point is $2^{-N+T(N)}$. The Hausdorff dimension is the limit of $\log(2^{-N+T(N)})/\log(2^{-N})$. The fact that $T(N)=o(N)$ guarantees that the Hausdorff dimension is 1.






