In this paper we consider the problem that how large the Hausdorff dimension of E ⊂ Rd needs to be to ensure the radii set of (d − 1)-dimensional spheres determined by E has positive Lebesgue measure. We obtain two results. First, by extending a general mechanism for studying Falconer-type problems in [4], we prove that it holds when dimH(E) > d− 1+ 1 d and in R2, the index 3 2 is sharp for thi...

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set E := { z : |z| ≤ r and |P (z)| ≤ n } , for a polynomial P of degree ≤ n. Usually, P is taken to be monic, and either Cartan’s Lemma or potential theory is used to estimate the size of E, in terms of Hausdorff contents, planar Lebesgue measure m2, or logarithmic...

We are familiar with the idea that a curve is a 1-dimensional object and a surface is 2-dimensional. What is the “dimension” of a set having infinite length but zero area? Similarly, the Cantor set has Lebesgue measure zero, but in some ways it is quite large (it is uncountable). How to quantify its size? In this talk we will answer these questions through the notions of Hausdorff measure and d...

Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R 2 with two independent approximation functions; that is if a certain sum converges then the set of all points (x, y) on the curve which satisfy simultaneously the inequalities qx < ψ1(q) and qy < ψ2(q) infinitely often has induced measure 0. This co...

Let C be a non–degenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ–approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. ...

The transfinite diameter is a way of quantifying the size of compact sets in Euclidean space. This quantity is related to the Hausdorff dimension and the Lebesgue measure, but gives a slightly different perspective on the set than either of those do. In this paper, we introduce the transfinite diameter, and outline some attempts to calculate this quantity for three sets in R. For z1, z2, . . . ...

We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in R2. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent syst...

We show that, given a set E ⊂ Rn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform