Juan Mat\'{i}as Sepulcre Mart\'{i}nez University of Alicante

Abstract : In this paper, we prove that infinite-dimensional vector spaces of $\alpha$-dense curves are generated by means of the functional equations $f(x)+f(2x)+\cdots+f(nx)=0$, with $n\geq 2$, which are related to the partial sums of the Riemann zeta function. These curves $\alpha$-densify a large class of compact sets of the plane for arbitrary small $\alpha$, extending the known result that this holds for the cases $n=2,3$. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the $n^{th}$ power of the density approaches the Jordan content of the compact set which the curve densifies.

Keywords : functional equations, space-filling curves, partial sums of the Riemann zeta function, alpha-dense curves, property of quadrature