Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

Abstract

This paper uses the zeta function methods to solve Falconer-type problems about sets of k-simplices whose endpoints belong simultaneously to a self-similar subset F of Z n (k n) and a disc B(x) of a large radius x. Assuming that the similarity transformations pairwise commute, we study four Euclidean metric invariants of the simplices, the most basic (and frequently studied) of which is the distance between endpoints of a 1-simplex. For each, we introduce a zeta function, derive its functional properties, and apply such information to derive a lower bound on the upper Minkowski dimension of F , which guarantees that the number of distinct metric invariants must be unbounded as x → ∞.