Distance and tube zeta functions of fractals and arbitrary compact sets

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N-dimensional Euclidean space RN, for any integer N≥1. It is defined by the Lebesgue integral ζA(s)=∫Aδd(x,A)s−Ndx, for all s∈C with Res sufficiently large, and we call it the distance zeta function of A. Here, d(x,A) denotes the Euclidean distance from x to A and Aδ is the δ-neighborhood of A, where δ is a fixed positive real number. We prove that the abscissa of absolute convergence of ζA is equal to dim‾BA, the upper box (or Minkowski) dimension of A. Particular attention is payed to the principal complex dimensions of A, defined as the set of poles of ζA located on the critical line {Res=dim‾BA}, provided ζA possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, ζ˜A(s)=∫0δts−N−1|At|dt, called the tube zeta function of A. Assuming that A is Minkowski measurable, we show that, under some mild conditions, the residue of ζ˜A computed at D=dimB⁡A (the box dimension of A) is equal to the Minkowski content of A. More generally, without assuming that A is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of A. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.