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On finitely many base $q$ expansions

Simon Baker, George Bender

TL;DR

This work provides the first explicit families of intervals in $(1,2)$ for which bases $q$ admit a nonempty set of points with exactly $m$ base-$q$ expansions and, moreover, positive Hausdorff dimension for all $m\ge3$. The authors develop a quantitative framework built from Cantor-set thickness, interleaving, and the Falconer–Yavicoli intersection theorem to locate these intervals around the $k$-Bonacci bases $q_k$, and they obtain explicit lower bounds on the dimension of the corresponding $\mathcal{U}_q^{(m)}$. A refined result for $\mathcal{B}_3$ uses a constructive set $A_q$ and Newhouse’s thickness-interleaving criterion to produce concrete neighbourhoods near certain $q_k$ and near $q_9$, with explicit Hausdorff-dimension guarantees. The methods bridge dynamical base-$q$ expansions, geometric measure theory, and fractal-intersection techniques, offering a robust path to quantify the size and location of multi-expansion bases in $(1,2)$.

Abstract

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions.

On finitely many base $q$ expansions

TL;DR

This work provides the first explicit families of intervals in for which bases admit a nonempty set of points with exactly base- expansions and, moreover, positive Hausdorff dimension for all . The authors develop a quantitative framework built from Cantor-set thickness, interleaving, and the Falconer–Yavicoli intersection theorem to locate these intervals around the -Bonacci bases , and they obtain explicit lower bounds on the dimension of the corresponding . A refined result for uses a constructive set and Newhouse’s thickness-interleaving criterion to produce concrete neighbourhoods near certain and near , with explicit Hausdorff-dimension guarantees. The methods bridge dynamical base- expansions, geometric measure theory, and fractal-intersection techniques, offering a robust path to quantify the size and location of multi-expansion bases in .

Abstract

Given some integer , we find the first explicit collection of countably many intervals in such that for any in one of these intervals, the set of points with exactly base expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of has positive Hausdorff dimension under certain conditions.
Paper Structure (15 sections, 30 theorems, 129 equations, 3 figures, 2 tables)

This paper contains 15 sections, 30 theorems, 129 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Thickness and interleaving
  4. A sufficient condition for $q \in \mathcal{B}_m$
  5. Proof of Theorem \ref{['theorem: main theorem']}
  6. Proof outline
  7. Construction of $P_i(q)$ and $Q_m(q)$
  8. The relative structure of the sets $P_i(q)$ and $Q_m(q)$
  9. Bounding $\beta_q$ below
  10. Proof of Theorem \ref{['theorem: main theorem']}
  11. Proof of Theorem \ref{['theorem: B_3 theorem']}
  12. Proof outline
  13. Construction of $A_q$
  14. Hausdorff distances and interleaving
  15. Proof of Theorem \ref{['theorem: B_3 theorem']}

Key Result

Theorem A

Let $m \in \mathbb{N}$ and let $k \geq K_m$ where If $|q - q_k| < q_k^{-(m+2)k-3}$, then $q \in \mathcal{B}_{m+2}$ and $\dim_{\mathrm{H}}(\mathcal{U}_q^{(m+2)}) \geq 1 - 1024(m+2)^{\frac{20}{19}}q^{4-k} > 0$.

Figures (3)

  • Figure 1:
  • Figure 2: The process of taking subsets of affine images of $\pi_q(S_{k-1})$ in order to bound $\beta_q$. For $|q - q_k| < q_k^{-2k-6}$, the relative structure of the sets in all cases where $q < q_k$, $q = q_k$ and $q > q_k$ are shown in Figure \ref{['figure: relative structure of P and Q sets']}.
  • Figure 3: The relative structure of the convex hulls of the sets $P_0(q) , \ldots P_m(q) , Q_m(q)$ and $B(q)$ when $|q - q_k| < q_k^{-(m+2)k-3}$, as is proved in Lemma \ref{['lemma: layout of endpoint of P and Q']}.

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem N
  • Theorem F&Y
  • Lemma 2.4
  • Lemma 2.5
  • ...and 37 more