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 ${mathbb R}^N$, for any integer $Nge1$. It is defined by $zeta_A(s)=int_{A_{delta}}d(x,A)^{s-N},mathrm{d} x$ for all $sinmathbb{C}$ with $operatorname{Re},s$ 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 $A_{delta}$ is the $delta$-neighborhood of $A$, where $delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $zeta_A$ is equal to $overlinedim_BA$, 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 $zeta_A$ located on the critical line ${mathop{mathrm{Re}} s=overlinedim_BA}$, provided $zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $tildezeta_A(s)=int_0^{delta} t^{s-N-1}|A_t|,mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $tildezeta_A$ computed at $D=dim_BA$ (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 Bakers theorem from the theory of transcendental numbers.
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