Tangential limits of potentials on homogeneous trees

Let T be a homogeneous tree of homogeneity q + 1. Let Delta denote the boundary of T, consisting of all infinite geodesics b = [b(0), b(1), b(2),...] beginning at the root, 0. For each b is an element of Delta, tau greater than or equal to 1, and a greater than or equal to 0 we define the approach region Omega(tau,a)(b) to be the set of all vertices t such that, for some j, t is a descendant of b(j) and the geodesic distance of t to b(j) is at most (tau - 1) j + a. If tau > 1, we view these as tangential approach regions to b with degree of tangency. We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition Sigma(T) f(p)(t)q(-gamma)\t\ < infinity, where p > 1 and 0 < gamma < 1, or p = 1 and 0 < gamma < 1. For 1 less than or equal to tau less than or equal to 1/gamma, we show that Gf (s) has limit zero as s approaches a boundary point b within Omega(tau,a)(b) except for a subset E of Delta of taugamma-dimensional Hausdorff measure 0, where H-taugamma (E) = sup(delta>0) inf{Sigma(i) q(-taugamma\ti\) : E a subset of the boundary points passing through t(i) for some i, \t(i)\ log(q) (1/delta)}.