LAUSR

In this paper we first prove analytical properties of zeta functions for discrete subsets of R that exhibit “self-similarity” with respect to an arbitrary finite set of (affine) similarities. We then show how such properties help solve Point Configuration resp. Sum-Product type problems over Z. We do so by first extending a classic one variable Tauberian theorem of Ingham to several variables to derive a non trivial lower bound on the average of coefficients of an appropriate multivariate zeta function. We then combine this with well known results from Diophantine Geometry that prove uniform bounds for the density of lattice points in families of algebraic hypersurfaces. Mathematics Subject Classifications: 11M41, 28A80, 52C10, 11J69, 11M32