Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$

Added later The answers below show that there exist closed sets $S$ of locally finite $ \mathcal{H}^1 $-measure whose tangent cone equals $\mathbf{R}^2$ on a subset of $S$ of positive measure. Can we produce a such closed set $ S$ with the additional property that $ S $ is the boundary of an open set? As above, if we remove the condition of local finitess of $ \mathcal{H}^1 $ a such example is again trivial and can be constructed using the graph of a highly oscillating function (e.g. $ \sin(1/x) $)

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$

Added later The answers below show that there exist closed sets $S$ of locally finite $ \mathcal{H}^1 $-measure whose tangent cone equals $\mathbf{R}^2$ on a subset of $S$ of positive measure. Can we produce a such closed set $ S$ with the additional property that $ S $ is the boundary of an open set? As above, if we remove the condition of local finitess of $ \mathcal{H}^1 $ a such example is again trivial and can be constructed using the graph of a highly oscillating function (e.g. $ \sin(1/x) $)

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the contingent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$

It seems reasonable that there exists an example of a closed set $ S \subseteq \mathbf{R}^2 $ such that $ \mathcal{H}^1(S) < \infty $ and $ T_S(x) = \mathbf{R}^2 $ for every $ x $ in a subset of $ S $ with positive $ \mathcal{H}^1 $ measure. Can anyone provide (or point out a reference for) a construction of such example?

Of course, if we replace the hypothesis of (locally) finiteness of the $ \mathcal{H}^1 $-measure with $ \sigma $-finiteness of the $ \mathcal{H}^1 $-measure then the construction is trivial. We can simply take $ S = (\{0\} \times \mathbf{R}) \cup \bigcup_{i=1}^\infty\big[(\{i^{-1}\} \times \mathbf{R}) \cup (\{-i^{-1}\} \times \mathbf{R})\big] $, for which evidently $$ T_S(x) = \mathbf{R}^2 \quad \textrm{for every $ x \in \{0\} \times \mathbf{R} $}.$$