Zeta Functions of Discrete Self-similar Sets and Applications to Point Configuration and Sum–Product-Type Problems

International audienceIn this paper we first prove analytical properties of zeta functions for discrete subsets of R^n 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 nontrivial 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