Zeta functions and solutions of Falconer‐type problems for self‐similar subsets of ℤn. Issue 5 (30th October 2015)

• Record Type: Journal Article
• Authors: Essouabri, Driss; Lichtin, Ben
• Is Part Of: Proceedings of the London Mathematical Society. Volume 111:Issue 5(2015:Nov.)
• Page Start: 1163
• Page End: 1205
• Publication Date: 2015-10-30
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pdv049
• Languages: English
• ISSNs: 0024-6115

Abstract

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 → ∞ .