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An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases

Jonathan Caalim, Nathaniel Nollen

TL;DR

The work analyzes Cantor real expansions in periodic alternate bases and seeks Solomyak-type bounds on nontrivial Galois conjugates of the base. It develops an $(\\alpha,\\beta)$-expansion framework, introduces semi-Parry/Parry notions, and constructs an analytic-analytic framework linking conjugates to zeros of parameterized power series, culminating in explicit bounds $|\\lambda| \\le 1/|M_i z(\\gamma;y)|$ under semi-Parry conditions. The authors prove an equality between the zero set of associated analytic functions and the closure of conjugates in the $\\gamma=0$, $\\beta>1$ regime (with a parallel treatment for $0<\\beta<1$), and provide irreducibility arguments to support the sharpness of the bounds. Special-case analysis for polynomial relations between the base elements yields additional explicit bounds and a catalog of results, thereby extending Solomyak-type conjugate bounds from classical beta-expansions to periodic Cantor real expansions in alternate bases.

Abstract

In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval $[γ, γ+1]$, where $γ\in \mathbb{R}$, with respect to an alternate base (i.e., a base which is a purely periodic sequence of real numbers). In particular, we study the case where the expansion of $γ+1$ is periodic. Under certain assumptions, the base satisfies algebraic properties. We compute the bounds for the norms of the nontrivial Galois conjugates associated with the base; thereby, extending the results of Solomyak on the classical beta expansions.

An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases

TL;DR

The work analyzes Cantor real expansions in periodic alternate bases and seeks Solomyak-type bounds on nontrivial Galois conjugates of the base. It develops an -expansion framework, introduces semi-Parry/Parry notions, and constructs an analytic-analytic framework linking conjugates to zeros of parameterized power series, culminating in explicit bounds under semi-Parry conditions. The authors prove an equality between the zero set of associated analytic functions and the closure of conjugates in the , regime (with a parallel treatment for ), and provide irreducibility arguments to support the sharpness of the bounds. Special-case analysis for polynomial relations between the base elements yields additional explicit bounds and a catalog of results, thereby extending Solomyak-type conjugate bounds from classical beta-expansions to periodic Cantor real expansions in alternate bases.

Abstract

In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval , where , with respect to an alternate base (i.e., a base which is a purely periodic sequence of real numbers). In particular, we study the case where the expansion of is periodic. Under certain assumptions, the base satisfies algebraic properties. We compute the bounds for the norms of the nontrivial Galois conjugates associated with the base; thereby, extending the results of Solomyak on the classical beta expansions.
Paper Structure (8 sections, 12 theorems, 66 equations, 1 figure, 1 table)

This paper contains 8 sections, 12 theorems, 66 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Periodic $B$-expansions
  3. The $(\alpha,\beta)$-expansion
  4. Structure of Conjugates of $\alpha$
  5. Proof of Theorem \ref{['equality']}
  6. Case 1: $\beta>1$
  7. Case 2: $0<\beta<1$
  8. Special Cases

Key Result

Proposition 2.1

Let $K_i:=\mathbb{Q}(\beta_1,\dots,\beta_{i-1},\beta_{i+1},\beta_n)$. Let $x\in K_i\cap [\gamma, \gamma+1]$. If $d(B;x)$ or $d^*(B;x)$ is eventually periodic such that $T^{i-1}(x)\neq0$, then $\beta_i$ is algebraic over $K_i$.

Figures (1)

  • Figure 1: Plot of $\mathscr{A}_R$ (red) and $\mathscr{C}_R$ (blue) for each given $R$

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Remark
  • Theorem 2.2
  • proof
  • Corollary 2.3: Main Result
  • Remark
  • Definition 3.1
  • Proposition 3.2
  • Remark
  • ...and 17 more