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**Definitions:**

Let $C \subset [0, 1]$ be a fat Cantor set, for parameter $0 < r < 1/3$. Thus intervals of width $r^n$ are removed from the middle of the previous intervals at each step. For the given range of $r$ this produces a fat cantor set, i.e. the measure of $C$ is nonzero. Write $C = \cap C_n$ for the usual decomposition of $C$, so that $C_n$ represents the leftover set after the $n$'th iteration, and define the address function $A: C \to [0, 1]$ by the following - $A(1) = 1, A(0) = 0$, and for $x \in C \cap (0, 1)$, the $k$'th binary decimal digit, $b_k A(x)$ of $A(x)$ is defined by the following:

$b_1 (x) = 0$ if $x$ lies to the left of the deleted interval in stage $1$, and $1$ otherwise.

For general $k$, $b_k A(x) = 0$ if $x$ lies to the left of the interval deleted from the connected component of $x$ in $C_{k-1}$ at stage $k$ and $1$ otherwise.

For each $x \in C$, define $$L_x (k) := sup \{{ j \in \mathbb Z^+ |\; b_{h+k-1} A(x) = b_k A(x) \;\textrm{for all}\; h \in  \mathbb Z^+ \; \textrm{with} \; h \leq j\}}$$

so that $L_x (k)$ is the length of the longest consecutive string of $0's$ or $1's$ beginning at the $k$'th decimal point of the address of $x$. By convention we shall always choose the binary expansion that ends in an infinite string of $1$’s when possible.

**Statement:**

Fix $r$, and consider the function $D: [0, 1] \to \mathbb R^{\geq0}$ defined by $D(x) := d(x, C)$, where $d$ denotes the Euclidean distance from a point to a set.

Then $D$ is Lipschitz continuous (with Lipschitz constant $1$), so is differentiable a.e. by Rademacher's theorem. I believe I have the following result:

**Theorem:** There exists a constant $M > 0$, depending only on $r$ such that $D$ is differentiable at $x \in C$ if and only if $\lim_{k \to \infty} k - ML_x (k) = \infty$.

There is also an explicit expression for $M$ in terms of $r$.

From a quick search I have not been able to find much on this topic. Has this result been proven before? And if so, is there any extension in the literature to the nonhomogeneous case, where a proportion $r_n$ is removed at the $n$'th stage?

Thanks in advance! Also I did not state some definitions in full, so feel free to ask for any clarification!

Sun, 28 Mar 2021 08:21 GMT