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Intermediate dimensions of complementary sets

Nicolas Angelini, Ursula Molter

TL;DR

This work analyzes the $\theta$-intermediate dimensions of complementary sets formed from a summable, nonincreasing gap sequence $a$ with sum $1$. It derives exact formulas for the $\theta$-intermediate dimensions of the countable set $D_a$ via box-dimension data, and, under the mild decay condition $\lim_{n\to\infty}(s_n/s_{n+1})^{1/n}=1$, determines $\overline{\dim}_{\theta} C_a$ (and its lower analogue) in terms of the Cantor-construction scales $s_n$ and the index function $\rho(n)$. The extremal $\theta$-intermediate dimensions of all complementary sets $E\in\mathcal{C}_a$ are shown to be bounded by those of $D_a$ and $C_a$, and under doubling-type assumptions on $a$ the attainable values form a closed interval with every intermediate value realized. The paper provides constructive methods to realize any target $t$ within the interval and connects these results to known bounds for Assouad and box dimensions, while posing open questions about the structure of attainable dimension functions for complementary sets.

Abstract

Given a positive, non-increasing sequence $a$ with finite sum equal to $1$, we consider the family of all closed subsets of $[0,1]$ whose complementary open intervals have lengths given by a rearrangement of the sequence $a$. We study the full range of possible $θ$-intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.

Intermediate dimensions of complementary sets

TL;DR

This work analyzes the -intermediate dimensions of complementary sets formed from a summable, nonincreasing gap sequence with sum . It derives exact formulas for the -intermediate dimensions of the countable set via box-dimension data, and, under the mild decay condition , determines (and its lower analogue) in terms of the Cantor-construction scales and the index function . The extremal -intermediate dimensions of all complementary sets are shown to be bounded by those of and , and under doubling-type assumptions on the attainable values form a closed interval with every intermediate value realized. The paper provides constructive methods to realize any target within the interval and connects these results to known bounds for Assouad and box dimensions, while posing open questions about the structure of attainable dimension functions for complementary sets.

Abstract

Given a positive, non-increasing sequence with finite sum equal to , we consider the family of all closed subsets of whose complementary open intervals have lengths given by a rearrangement of the sequence . We study the full range of possible -intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.

Paper Structure

This paper contains 7 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Complementary Sets
  5. $\theta-$intermediate dimension of $C_a$ and $D_a$
  6. Maximum and minimum dimensions
  7. Attainable values

Key Result

Proposition 2.3

If $\dim_L E<\dim_A E$, then for all $\theta \in (0,1]$, The same holds replacing $\overline{\hbox{\rm dim}}_{\,\theta}$ and $\overline{\hbox{\rm dim}}_{\rm B}\,$ with $\underline{\hbox{\rm dim}}_{\,\theta}$ and $\underline{\hbox{\rm dim}}_{\rm B}\,$ everywhere.

Theorems & Definitions (17)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Corollary 2.8 Banaji-R22
  • Proposition 3.1
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Remark 5.1
  • ...and 7 more