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Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

Let ${\mathrm{\Omega <\#comment/>}}^{+}\subset <\#comment/>{\mathbb{R}}^{n+1}$be a bounded $\mathrm{\delta <\#comment/>}$-Reifenberg flat domain, with $\mathrm{\delta <\#comment/>}>0$small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}={\mathbb{R}}^{n+1}\setminus <\#comment/>\stackrel{¯<\#comment/>}{{\mathrm{\Omega <\#comment/>}}^{+}}$is an NTA (non-tangentially accessible) domain as well and denote by ${\mathrm{\omega <\#comment/>}}^{+}$and ${\mathrm{\omega <\#comment/>}}^{-<\#comment/>}$the respective harmonic measures of ${\mathrm{\Omega <\#comment/>}}^{+}$and ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}$with poles $p^{\pm <\#comment/>}\in <\#comment/>{\mathrm{\Omega <\#comment/>}}^{\pm <\#comment/>}$. In this paper we show that the condition that $\log <\#comment/>\frac{d{\mathrm{\omega <\#comment/>}}^{-<\#comment/>}}{d{\mathrm{\omega <\#comment/>}}^{+}}\in <\#comment/>\mathrm{VMO}<\#comment/>\left({\mathrm{\omega <\#comment/>}}^{+}\right)$is equivalent to ${\mathrm{\Omega <\#comment/>}}^{+}$being a chord-arc domain with inner unit normal belonging to $\mathrm{VMO}<\#comment/>\left({\mathcal{H}}^n{|}_{\mathrm{\partial <\#comment/>}{\mathrm{\Omega <\#comment/>}}^{+}}\right)$.