Zeta functions and solutions of Falconer‐type problems for self‐similar subsets of ℤn. Issue 5 (30th October 2015) - Essouabri, Driss; Lichtin, Ben
Abstract : This paper uses the zeta function methods to solve Falconer‐type problems about sets ofk ‐simplices whose endpoints belong simultaneously to a self‐similar subsetF ofZ n ( k ⩽ n ) and a discB ( x ) of a large radiusx . 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 ofF, which guarantees that the number of distinct metric invariants must be unbounded asx → ∞ .