Integer Cantor Sets: Arithmetic Combinatorial Properties

TL;DR

This paper explores the arithmetic combinatorial properties of integer Cantor sets, providing new examples, characterizations, and theorems related to their distribution, ergodic behavior, and pair correlation.

Contribution

It introduces novel examples of Cantor sets with intersective properties, characterizes uniform distribution, and proves polynomial mean ergodic theorems for these sets.

Findings

Examples of Cantor sets with intersective property and power savings

Characterization of uniform distribution of these sets

Establishment of polynomial mean ergodic theorems and metric pair correlation results

Abstract

Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective property with power savings (b) characterize uniform distribution, (c) establish polynomial mean ergodic theorems and (d) study metric pair correlation of Cantor sets.