We associate to each stable Higgs pair $(A_0,{\Phi }_0)$ on a compact Riemann surface $X$ a singular limiting configuration $(A_{\infty },{\Phi }_{\infty })$, assuming that $det\Phi$ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions $(A_t,t{\Phi }_t)$ to Hitchin’s equations, which converge to this limiting configuration as $t\to \infty$. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

We prove that the set of singular vectors in ${\mathbb{R}}^d$, $d\ge 2$, has Hausdorff dimension $\frac{d^2}{d+1}$ and that the Hausdorff dimension of the set of $\epsilon$-Dirichlet improvable vectors in ${\mathbb{R}}^d$ is roughly $\frac{d^2}{d+1}$ plus a power of $\epsilon$ between $\frac{d}{2}$ and $d$. As a corollary, the set of divergent trajectories of the flow by $diag(e^t,\dots ,e^t,e^{-dt})$ acting on ${SL}_{d+1}\left(\mathbb{R}\right)/{SL}_{d+1}\left(\mathbb{Z}\right)$ has Hausdorff codimension $\frac{d}{d+1}$. These results extend the work of the first author.

Let $E\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega :={\mathbb{R}}^{n+1}\setminus E$ satisfy Carleson measure estimates and are $\epsilon$-approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure.