arXiv subjects
Publications and source records attributed to Burton Randol.
This paper describes, in the case of the unit circle, several applications of a geometrically intrinsic treatment of counterparts of classical electrostatics, previously developed in [4] and [5].
Suppose $D$ is a suitably admissible compact subset of $\mathbb{R}^k$ having a smooth boundary with possible zones of zero curvature. Let \mbox{$R(T,θ,x)= N(T,θ,x) - T^{k}\mathrm{vol}(D)$,} where $N(T,θ,x)$ is the number of integral lattice points contained in an $x$-translation of $Tθ(D)$, with $T >0$ a dilation parameter and $θ\in SO(k)$. Then $R(T,θ,x)$ can be regarded as a function with parameter $T$ on the space $E_{*}^{+}(k)$, where $E_{*}^{+}(k)$ is the quotient of the direct Euclidean group by the subgroup of integral translations, and $E_{*}^{+}(k)$ has a normalized invariant measure which is the product of normalized measures on $SO(k)$ and the $k$-torus. We derive an integral estimate, valid for almost all $(θ,x) \in E_{*}^{+}(k)$, one consequence of which in two dimensions is that for almost all $(θ,x) \in E_{*}^{+}(2)$, a counterpart of the Hardy circle estimate \mbox{$(1/T)\int_{1}^{T} |R(t,θ,x)\,dt| \ll T^{\frac{1}{4} +ε}\;$}is valid with an improved estimate. We conclude with an account of hyperbolic versions for which, drawing on previous work of Hill and Parnovski \cite{hill-parnovski}, we give counterparts in all dimensions, for both the compact and non-compact finite volume cases.
This is an expanded version of [arXiv:1107.4836v1 [math.DS]]. Using techniques from [Chapter XI, The Selberg Trace Formula, in Eigenvalues in Riemannian Geometry, by Isaac Chavel], in which a differential-geometrically intrinsic treatment of counterparts of classical electrostatics was introduced, it is shown that on some compact manifolds, certain stable configurations of points which mutually repel along all interconnecting geodesics become equidistributed as the number of points increases.
It is shown that on compact hyperbolic manifolds, certain stable configurations of points which mutually repel along all interconnecting geodesics become equidistributed as the number of points increases
We study the asymptotic behavior of discrete measures on S^{n-1} that are induced by radially projecting point masses concentrated on the integral lattice-points within dilates of a compact body D, for various classes of D. The results depend sensitively on the differential geometric properties of the boundary of D.