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Integer Cantor Sets: Arithmetic Combinatorial Properties

Alex Burgin, Anastasios Fragkos, Michael T. Lacey, Dario Mena, Maria Carmen Reguera

TL;DR

The paper investigates arithmetic-ergodic properties of classical integer Cantor sets $\mathcal{C}_{b,D}$, introducing and analyzing the digit-structure-driven phenomena that govern distribution and recurrence. It proves polynomial uniform distribution for pairs $(k_n, s_b(k_n))$ on $[0,1]^2$, studies modular (mod $q$) distributions of $k_n$ and $s_b(k_n)$, and establishes mean ergodic theorems along Cantor sequences. By connecting digit-set geometry to additive-energy conditions, it provides criteria for van der Corput and intersective properties with power savings and derives sufficient conditions for Poissonian pair correlations for almost all shifts $\alpha$. The results highlight a rich interplay between combinatorial digit constraints, modular distribution, and ergodic-theoretic limits, with implications for recurrence and distribution along sparse digit-restricted sets.

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.

Integer Cantor Sets: Arithmetic Combinatorial Properties

TL;DR

The paper investigates arithmetic-ergodic properties of classical integer Cantor sets , introducing and analyzing the digit-structure-driven phenomena that govern distribution and recurrence. It proves polynomial uniform distribution for pairs on , studies modular (mod ) distributions of and , and establishes mean ergodic theorems along Cantor sequences. By connecting digit-set geometry to additive-energy conditions, it provides criteria for van der Corput and intersective properties with power savings and derives sufficient conditions for Poissonian pair correlations for almost all shifts . The results highlight a rich interplay between combinatorial digit constraints, modular distribution, and ergodic-theoretic limits, with implications for recurrence and distribution along sparse digit-restricted sets.

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.
Paper Structure (10 sections, 17 theorems, 80 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Measure Preserving Systems
  5. Cantor Sets
  6. Intersective and van der Corput Properties
  7. Uniform Distribution on [0,1]²
  8. Uniform Distribution on Za x Za'
  9. Ergodic Theorems
  10. Metric Pair Correlation

Key Result

Lemma 2.3

Let $a_1, a_2, \ldots, a_n$ be elements of a Hilbert space $H$, of norm at most $1$. For all integers $1\leq H \leq N$, we have

Figures (1)

  • Figure 1: An illustration of the set $E_4$ and $E_5$ above the axis and $E_6$ on the axis. Note that additional base points are added as $E_k$ increases.

Theorems & Definitions (41)

  • Remark 1.2
  • Lemma 2.3: van der Corput Inequality
  • Proposition 2.5
  • proof
  • Lemma 2.9
  • Definition 3.1
  • Definition 3.3
  • Proposition 3.6
  • proof
  • Proposition 3.8
  • ...and 31 more